v15n1

____ THE ______ MATHEMATICS ___ _________ EDUCATOR _____ Volume 15 Number 1

Spring 2005

MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA

Editorial Staff

A Note from the Editor

Editor Ginger A. Rhodes

Dear TME Readers,

Associate Editors Erin Horst Margaret Sloan Erik Tillema

I would like to present the first of two issues of Volume 15 of The Mathematics Educator (TME). In keeping with the tradition of TME, this issue offers an array of topics ranging from current debates in mathematics education to the role of mathematics teaching in developing a socially just society. We hope you find these articles both educational and stimulating.

Mark W. Ellis and Robert Q. Berry III as well as Serkan Hekimoglu and Margaret Sloan’s articles discuss historical and current debates that revolve around mathematics education. Ellis and Berry examine past movements in mathematics education and describe the current debates as a result of a fundamental conflict of paradigms. They propose that the current reform movement represents a paradigm shift that has the potential to transform the ways that students experience success in school mathematics. Hekimoglu and Sloan examine opposing viewpoints on the National Council of Mathematics Standards and discuss ways that a shared vision for the future of mathematics education could be developed.

Two research studies are presented. First, Regina Mistretta describes a research study that examined effects that training sessions have on pre-service teachers’ ability to evaluate and integrate instructional technology into the mathematics classroom. The second research report, by Ayhan Kursat Erbas, begins by describing the role of algebra in the Turkish educational system. He then examines three variables in order to predict algebra performance of ninth grade students in Turkey.

In the opening editorial Deborah Loewenberg Ball, Imani Masters Goffney, and Hyman Bass highlight three ways in which mathematics teaching can build a socially just and diverse democracy. They call our attention to the need to significantly change teachers’ education and professional development as well as a need for a diverse teaching force in order to meet the needs of the diverse students in mathematics classrooms.

Along with the editorial staff, authors in this issue have worked diligently and thoughtfully to present their work to the larger mathematics education community. I have given you a glimpse into the issues discussed by the authors and hope to have intrigued you to read further.

I would also like to bring your attention to the work of TME. Through the promotional efforts of our editorial staff over the last year we have increased our readership as well as the number of article submissions. It is TME’s purpose to encourage the interchange of ideas among the mathematics education community. I would like to invite and encourage TME readers to contribute to our purpose by reviewing manuscripts, submitting manuscripts, visiting our website, or joining our editorial staff (see pp. 46–47). As a final note, I would like to thank the numerous people whose work makes the success of TME possible, including reviewers, authors, faculty, and, especially, the editorial staff.

Publication Stephen Bismarck Dennis Hembree Advisors Denise S. Mewborn Nicholas Oppong James W. Wilson

MESA Officers 2004-2005 President Zelha Tunc-Pekkan Vice-President D. Natasha Brewley Secretary Amy J. Hackenberg Treasurer Ginger A. Rhodes NCTM Representative Angel Abney Undergraduate Representative Erin Cain Jessica Ivey Erin Bernstein

Ginger A. Rhodes 105 Aderhold Hall The University of Georgia Athens, GA 30602-7124

tme@uga.edu www.coe.uga.edu/tme

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 2005

Volume 15 Number 1

Table of Contents 2

Guest Editorial… The Role of Mathematics Instruction in Building a Socially Just and Diverse Democracy DEBORAH LOEWENBERG BALL, IMANI MASTERS GOFFNEY, & HYMAN BASS

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The Paradigm Shift in Mathematics Education: Explanations and Implications of Reforming Conceptions of Teaching and Learning MARK W. ELLIS & ROBERT Q. BERRY III

18 Integrating Technology Into the Mathematics Classroom: The Role of Teacher Preparation Programs REGINA M. MISTRETTA 25 Predicting Turkish Ninth Grade Students’ Algebra Performance AYHAN KURSAT ERBAS 34 A Compendium of Views on the NCTM Standards SERKAN HEKIMOGLU & MARGARET SLOAN 44 48 45 46

Upcoming conferences Subscription form Submissions information About the cover © 2005 Mathematics Education Student Association All Rights Reserved

The Mathematics Educator 2005, Vol. 15, No. 1, 2–6

Guest Editorial…

The Role of Mathematics Instruction in Building a Socially Just and Diverse Democracy Deborah Loewenberg Ball Imani Masters Goffney Hyman Bass As elementary school teachers, Deborah and Imani did not just teach academic subjects. They taught their pupils skills and knowledge to help develop them as individuals and as members of a collective. Subject matters offered important resources for these social goals: they and their students read literature in the voices of a wide range of people, about experiences both similar to and different from theirs. They studied other cultures and learned about work, life, and practice in a variety of societies and settings. And they learned that issues of voice, experience, culture, and setting were important threads in the tapestry of what it means to be human. The work they did with their pupils across these academic subjects was, of course, also aimed at developing the children’s skills and knowledge, their capacity to interpret texts and artifacts, to reason in disciplined ways, and to solve problems within and beyond these domains.

Their young students were also resources for the goals toward which Deborah and Imani worked as teachers. From a variety of cultural backgrounds, and a Deborah Loewenberg Ball is the William H. Payne Collegiate Professor of Mathematics Education and Teacher Education, and Director of Teacher Education, at the University of Michigan.

Ball's work focuses on studies of instruction and the processes of learning to teach. She also directs several research projects that investigate efforts to improve teaching through policy, reform initiatives, and teacher education..

Imani Masters Goffney is a doctoral student in mathematics and teacher education at the University of Michigan. She earned her bachelor's degree from Spelman College in 2000 and her Masters' degree in Curriculum Development from the University of Michigan in April 2002. Her research interests include innovations in teacher education and mathematics education, both at the pre-service level and in professional development, and issues of equity.

Hyman Bass is the Roger Lyndon Collegiate Professor of Mathematics and Mathematics Education at the University of Michigan. He is President of the International Commission on Mathematics Instruction. During the past seven years he has been collaborating with Deborah Ball and her research group at the University of Michigan on the mathematical knowledge and resources entailed in the teaching of mathematics at the elementary level.

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wide range of communities, their students thought differently from one another, and they brought ideas and experiences to offer to the collective work in their classes. They were children — they made friends, argued and fought, and were generously caring. What they did, said, and felt comprised their classroom’s working environment and offered a myriad of opportunities for learning. Over time, Deborah and Imani each learned to listen to and notice what the children brought and to use and mediate their differences.

All this was well and good — in reading, social studies, art, music, and even science. But mathematics seemed isolated from the rest. There seemed little to discuss, little opportunity to notice and use the diversity of their students. Deborah and Imani explained ideas and procedures, the students practiced, and they all reviewed. Although the two teachers thought the students were capable mathematically, the students did not think so. Some viewed themselves as “good at math,” while others disparaged their own abilities. Deborah and Imani saw differences in accomplishment produced from their instruction, and they worried. They grew concerned about which students were coming to see themselves as “bad at math,” and were quite sure that the source lay not with these students, but in their teaching.

Now, when the three of us discuss these teaching experiences, we understand a different landscape than Deborah and Imani knew to see then. We recognize that mathematics — and the ways in which teachers teach it — is a key resource for building a socially just and diverse democracy. While other school subjects, too, offer resources for democratic education and social justice, mathematics makes its own unique contributions to these goals. Instead of seeing mathematics as culturally neutral, politically irrelevant, and mainly a matter of innate ability, we see it as a critical lever for social and educational progress (Moses & Cobb Jr., 2001) if taught in ways that make Building a Socially Just and Diverse Democracy

use of its special resources. Three main points structure the perspective that we take in this essay.

First, in order to enable all students to be successful with mathematics, we see that some elements of “good teaching” of mathematics — listening closely to students’ ideas, for example, or being sensitively careful at the interface between mathematical and everyday language — are especially important. They are important in order to recruit students into mathematics, as well as to help them succeed there. Consider the imperative to listen closely to students, and to be sensitive to the boundaries between mathematical and everyday language: Students who are working on mathematics in an English that they are just learning1 express mathematical ideas in ways that seemed to us unusual or hard to understand at times; but when Deborah and Imani focused carefully, they heard significant mathematical insights they had previously missed, or misunderstood. The many varieties of English spoken in the classroom make it especially important to notice the ambiguities between technical and everyday uses of English: For example, what does it mean for a number to be “odd,” or “big”? What is a “right” angle? What about “similar” figures, or “equivalent” fractions? Why are some numbers “rational” and others not, and still others “radicals?” Is there a synonym for “regular,” and what is the distinctive technical meaning of a “group” — in elementary school or higher level math?

All these are words used one way in everyday talk, and in other ways in mathematics. Mathematics often uses and specializes everyday language, sometimes metaphorically, rather than coining a separate technical vocabulary (Halliday, 1978), thus both enabling and complicating entry to its register (Pimm, 1987).

Teachers also coin expressions to support students’ learning, saying that a number “goes into” another, or that one “borrows” from the tens. So although listening closely and being careful about the differences between technical and everyday uses of mathematical language are important aspects of “good teaching,” they demand emphasis in order to make mathematical success both common and expected.

Second, the disparities in mathematics achievement are tightly coupled with social class and race, and have not narrowed over the last decade despite a rhetoric of “mathematics for all.” Some have come to suspect that some aspects of “good teaching” may unwittingly create, reproduce, or extend inequities among students, differences deeply rooted in the inequalities of our society (Ball, Hoover, Lewis, Bass, & Wall, 2003). Take an example: A glance at Deborah Loewenberg Ball, Imani Masters Goffney, & Hymann Bass

mathematics textbooks, even those newly designed or revised, reveals the settings for many mathematics problems to be most familiar to middle class white students. Plans for garden plots, mileage covered on family vacations, stereotypical images of “family,” allowance plans — these and other “meaningful” and “real world” contexts may be more familiar and engaging to some students than to others. The effort to wade through an unfamiliar context in order to get to the mathematics can impede students’ learning (Lubienski, 2002). The enthusiasm for “real world” problems, left unchecked, may disadvantage students for whom the chosen settings are not understood or valued. This is not to say that problems or contexts may not be useful to students, only that contexts are often social or cultural and depend for their usefulness on students’ experiences. Attentive to this, some educators work to design contexts that are rooted in broader and more diverse experience and culture. They might use African designs as a site for studying geometric patterns, or urban street games as settings for using complex numerical strategies. Still, the difficulties that can arise from uneven familiarity with particular “contexts” require vigilance. We return below to the rich possibilities inherent in the use of cultural contexts.

Other practices of teaching thought to be “good” also deserve closer scrutiny — reluctance to “tell” students or to be explicit, for example. Letting students figure out crucial mathematical practices — how to compare representations, or how to build a mathematical argument — for themselves may well mean that only some students figure them out. This is not benign: Past evidence suggests that white or Asian middle class students, often male, tend to learn these implicitly, while many others do not (RAND Mathematics Study Panel, 2003). The contemporary enthusiasm for instructional approaches in which the teacher “facilitates” and refrains from being direct may be more congruent with some students’ experiences and practices than others, thus inadvertently advantaging those students if participation in such discourse is not explicitly taught (Delpit, 1988; Heath, 19xx; Heath, 1983; Lubienski, 2002). And, moreover, explicit guidance for learning complex skills or ideas is essential if all students are to develop such capacities.

Leaving the construction of these skills to chance can make student success susceptible to cultural differences in discursive norms.

Affirming students’ accomplishments, rewarding success, and praise are all ways in which good teachers encourage and inspire students to work hard and to see 3

themselves as mathematically capable (Boaler & NetLibrary Inc., 2000; Cohen, Lotan, Abram, Scarloss, & Schultz, 2002). But these also signal to students what it means to be “good at math.” Unexamined, these messages may communicate a narrow perspective on what mathematical ability is, and thus assign competence unevenly and without attention to a full range of mathematical skill and practice, and their diverse forms of expression.

So far we have discussed what may underlie significant and persistent disparities in mathematics achievement, efficacy, and success. Although many important societal factors shape these disparities, instructional practices also matter. Instruction can take aim at pervasive inequality, or it can reinforce or even create it. Too often, unexamined, it may do the latter.

Thus, learning to examine who and what is being valued and developed in math class is essential.

Still, our argument would be incomplete if we did not also consider what mathematics — and mathematics instruction — can contribute to education for democracy. As Malloy (2002) argues, mathematics education that is oriented to promote democratic goals can “provide students with an avenue through which they can learn substantial mathematics and can [develop into] productive and active citizens” (p. 21, emphasis added). Clearly, we need vigorous efforts to improve every student’s access to and development of usable mathematical literacy, including the skills for everyday life, preparation for the increasing mathematical demands of even relatively non-technical workplaces, and for continued mathematical study.

The need for collective commitment to this goal has never been greater. In addition, however, we claim that mathematics has a special role to play in educating young people for participation in a pluralistic democratic society. Making that possible depends on instruction that uses the special resources that mathematics holds for realizing these broader societal aims.

One way in which mathematics teaching can help to build the resources for a pluralistic society is through the development of tools for analysis and social change. Mathematics offers tools to examine and analyze critically the deep economic, political, and social inequalities in our society, for studying crucial societal problems, and for considering a host of issues that can be understood and critiqued using quantitative tools. For example, who voted in the last election and why? How does the Electoral College shape whose votes count most in a presidential election? How do our income and inheritance tax laws shape the 4

distribution of wealth and access to fundamental resources, as well as what is valued? How does our system of school funding, for example through real estate taxes, shape the quality of education that different children in our country receive? Developing and using the mathematical skills that enable young people to engage in social analysis and improvement is one way in which mathematics can contribute to the development of a diverse democracy.

A second way in which mathematics teaching can play a role in education for democracy is as a setting for developing cultural knowledge and appreciation, important resources for constructive participation in a diverse society. Mathematics represents an ancient and remarkable set of cultural achievements and engagements. As such, the historical development of mathematical ideas and methods offers a medium for studying history and culture and their intersections in domains of human activity as diverse as architecture, art, music, science, and religion. Mathematics offers opportunities for young people to learn about their own cultural heritage and that of others. Such learning is crucial for developing the understanding and appreciation of diverse traditions, values, and contributions, and ways to notice, respond to, and use them. Such learning is also crucial for developing a sense of one’s own cultural identity, and sense of self and membership, both for oneself and also as a participant in the broader cultural milieu.

But a third way that mathematics teaching can support the development of democratic goals — the one on which we focus here — is through the skills and norms embedded in mathematical practice itself. In other words, we argue that it is not just the content of mathematics and its tools that contributes to democratic goals, but the very nature of mathematical work. Mathematics instruction, we claim, can offer a special kind of shared experience with understanding, respecting, and using difference for productive collective work. How so? Consider that mathematics is centrally about problem solving, and about discovering and proving what is true. Alternative interpretations and representations of a problem can often serve to open a path to its solution; sometimes a novel metaphor, diagram, or context can crack a difficult part of a problem. At the same time, the use of difference is structured and supported by common disciplinary language, norms, and practices. Terms must be precisely defined and used in common ways.

Disagreements are resolved not by shouting or by plurality, but by reasoned arguments whose construction can be taught and learned. Decisions such Building a Socially Just and Diverse Democracy

as whether 0 is even or odd, or how to interpret the 3 5 4 meaning of , whether is greater or less than , or 4

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whether a solution to a particular problem is valid are subject to mathematical reasoning, not governed by desire! or power. !Moreover, mathematical !reasoning is a practice to be learned, not an innate talent.

In these ways, mathematics instruction can deliberately help young people learn the value of others’ perspectives and ideas, as well as how to engage in and reconcile disagreements. Mathematics instruction can be designed to help students learn that differences can be valuable in joint work, and that diversity in experience, language, and culture can enrich and strengthen collective capacity and effectiveness. Students can also learn that mathematics is not an arena in which differences are resolved by voting. Politics is an arena in which differences are managed in this way, but the study of literature or mathematics is not. In a democratic society, how disagreements are reconciled is crucial. But mathematics offers one set of experiences and norms for doing so, and other academic studies and experiences provide others. In literature, differences of interpretation need not be reconciled, in mathematics common consensus matters. In this way, mathematics contributes to young people’s capacity for participation in a diverse society in which conflicts and are not only an inescapable part of life, but their resolution, in disciplined ways, is a major source of growing new knowledge and practice.

How might instruction be designed to serve both mathematical and democratic ends? One element would lie with the mathematical tasks selected. Tasks that serve to develop common skills, language, and practices offer ways that can help to build the common skills needed for class work on mathematics. Also useful are tasks that yield to alternative representations, so that students’ understanding of the material is deepened through the different ways in which their classmates see the ideas. Although it is valuable to use mathematical tasks that profit from others’ interpretations, such tasks should not, however, depend unfairly on unevenly distributed cultural experience or knowledge.

How mathematical tasks are used is crucial in determining whether or not their potential is realized in classrooms. If not carefully structured and guided, cognitively complex tasks can degrade to simple routine problems, and problems ripe with opportunity for reasoning and representation can become algorithmic (Stein, 1996). Similar vigilance is needed in order for tasks to serve as contexts for the Deborah Loewenberg Ball, Imani Masters Goffney, & Hymann Bass

development of democratic skills and dispositions.

Such vigilance is centered on cultivating attention to and respect for others’ mathematical ideas. Students would need to develop a consistent stance of civility with one another, a stance based on intellectual interest and respect, not mere social politeness or “niceness.”

This would require learning to listen carefully to others’ ideas, and checking for understanding before disagreeing. Other skills, norms, and practices of collective mathematical work include giving credit to others’ ideas — referring to ideas by their authors’ names, for example — and critiquing ideas, not people, using the tools and practices of the discipline. Students would work to seek agreement on meanings and solutions, drawing on past shared experiences, definitions, ideas, and agreements about meaning, and they would use and contribute to one another’s ideas in a collective effort to solve and understand the mathematics and the problems on which they are working. Important to our argument is that these skills and practices that are central to mathematical work are ones that can contribute to the cultivation of skills, habits, and dispositions for participation in a diverse democracy.

For mathematics instruction to contribute to the building of a socially just and diverse democracy will require more than care with curriculum and teaching. It will also require more than committed teachers, sensitive to and skillful in working toward these aims (Ladson-Billings, 2001). Accomplishing this would require significant change in teachers’ education and professional development, no small task. But who these teachers are matters as well. We need a teaching force diverse in race, culture and ethnicity, and linguistic resources. The current teaching population is disproportionately white, female, and middle-class.

The profession responsible for teaching our nation’s children should include people of a wider range of cultural and experiential resources, both because young learners should have access to more diversity in the teachers from whom they learn (Irvine, 2003), and because the collective knowledge, practice, and norms of the profession would be improved if its members were more diverse. Responsible for helping prepare young people for life in society, teachers — and the mathematics instruction they offer — must collectively represent and take advantage of the multicultural nature of that society for individual and common good.

REFERENCES Adler, J. (2001). Teaching mathematics in multilingual classrooms.

Boston: Kluwer academic publishers.

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Ball, D. L., Hoover, M., Lewis, J., Bass, H., & Wall, E. (2003). In attention to equity in teaching elementary school mathematics.

Prepared in Draft form for 2003 annual meeting of the AERA. Baugh, J. (1999). Out of the mouths of slaves: African american language and educational malpractice. Austin: The University of Texas Press.

Malloy, C. E. (2002). Democratic access to mathematics through democratic education. In L. D. English, (Ed.), Handbook of International Research in Mathematics Education, (pp. 17– 25). Mahwah, NJ: Lawrence Erlbaum Associates.

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

Boaler, J., & NetLibrary Inc. (2000). Multiple perspectives on mathematics teaching and learning. Westport, Conn.: Ablex.

Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms. London: Routledge.

Cohen, E. G., Lotan, R. A., Abram, P. L., Scarloss, B. A., & Schultz, S. E. (2002). Can groups learn? Teachers College Record 104 (6), 1045–1068.

RAND Mathematics Study Panel, D. L. B., Chair. (2003).

Mathematical proficiency for all students: Toward a strategic research and developmenet program in mathematics education. Arlington: RAND.

Delpit, L. D. (1988). The silenced dialogue: Power and pedagogy in educating other people's children. Harvard Educational Review, 58 (3), 280–297.

Halliday, M. A. K. (1978). Language as a social semiotic.

Baltimore, MD: University Park Press.

Heath, S. B. (1983). Ways with words: Language, life and work in communities and classrooms. New York: Cambridge University Press.

Irvine, J. J. (2003). Educating teachers for diversity: Seeing with a cultural eye. New York: Teachers College Press.

Ladson-Billings, G. (2001). Crossing over to canaan: The journey of new teachers in diverse classrooms (1st ed.). San Francisco: Jossey-Bass.

Lubienski, S. T. (2002). Research, reform, and equity in U.S. Mathematics education. Mathematical Thinking and Learning, 4(2–3), 103–125.

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Schleppegrell, M. J. (2002). Challenges of the science register for esl students: Errors and meaning-making. In Developing advanced literacy in first and second languages (pp. 119– 142). Mahwah, NJ: Lawrence Erlbaum Associates.

Stein, M. K., Grover, B.W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms.

American Educational Research Journal, 33(2), 455–488.

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We include here those students whose mother language is another world language as well as those who speak one of many dialects of the English language (Adler, 2001; Baugh, 1999; Schleppegrell, 2002)

Building a Socially Just and Diverse Democracy

The Mathematics Educator 2005, Vol. 15, No. 1, 7–17

The Paradigm Shift in Mathematics Education: Explanations and Implications of Reforming Conceptions of Teaching and Learning Mark W. Ellis Robert Q. Berry III In this article, we argue that the debates about mathematics education that have arisen in the United States over the past decade are the result of a major shift in how we conceptualize mathematical knowledge and mathematics learning. First, we examine past efforts to change mathematics education and argue they are grounded in a common traditional paradigm. Next, we describe the emergence of a new paradigm that has grown out of a coalescence of theories from cognitive psychology, an awareness of the importance of culture to learning, and the belief that all students can and should learn meaningful mathematics. Reforms grounded in the new paradigm have the potential to dramatically alter the way in which students—as well as which students— experience success in school mathematics. We discuss some implications of these reforms related to how mathematics educators might work with teachers of mathematics.

An examination of articles and reports about what should be done to improve mathematics education in the United States can be alarming. One finds calls for more skill building and less use of calculators while at the same time there is a push to teach for understanding through the use of technology; there are textbooks deemed “superior” that require teachers to take students individually in lock-step fashion through a vast collection of skills and procedures, while another set of “excellent” texts position teachers as guides for cooperative learning activities through which students construct an understanding of a few key concepts (American Association for the Advancement of Science, 2000; Clopton, McKeown, McKeown, & Clopton, 1999). Some want more standardized testing of mathematical skills, while others want more authentic assessments that are based on mathematical standards. One perspective holds tracking in mathematics to be inevitable while the other sees it as inequitable. Yet, everyone claims to want what is Mark Ellis is a doctoral candidate in Education at the University of North Carolina Chapel Hill and expects to graduate this year.

He taught mathematics in California public schools for six years and developed interests in issues of pedagogy and assessment as they relate to equity in mathematics education. In August 2005 he will join the faculty in the College of Education at California State University Fullerton where he will run the middle level mathematics teacher preparation program.

Dr. Robert Q. Berry, III is an assistant professor of mathematics education at the University of Virginia. His research interests focus on equity issues in preK-12 mathematics education and computational estimation. He received his Doctor of Philosophy (Ph.D.) degree from the University of North Carolina at Chapel Hill in May 2003.

Mark Ellis & Robert Q. Berry III

“best” for all students. How can this be?

In this article, we develop a critical view of efforts to improve mathematics education over the past century which allows us to demonstrate that the current debates can be understood as the result of a fundamental conflict of paradigms. We argue that up until the 1980s, efforts at change came out of a common perspective or paradigm in which mathematics was viewed as a disembodied set of objective truths to be communicated to students, most of whom would then struggle to internalize them.

Thorndike’s Stimulus-Response Bond theory, the progressive movement’s efforts to make learning vocationally relevant and schooling more efficient, the curricular changes associated with New Math, and the emphasis on rote facts, skills, and procedures of the back-to-basics movement all developed within this common paradigm. We will discuss these efforts at improving mathematics education and provide some perspective on their failure to positively impact the inequitable outcomes in mathematics learning that have persisted since the early twentieth century.

A History of Revisions in Mathematics Education Revision can be characterized as a renewal effort that captures educators’ attention for a short period of time but fails to address critical issues that are at the root of students’ difficulties with mathematics.

Revision perpetuates a “quick fix” approach whereby new components are adapted to fit within the bounds of the accepted paradigm. Thus, revision in mathematics education leads to surface level modifications but does 7

little to substantively alter deeply held beliefs about the nature of mathematics, how it is to be taught, the sort of learning that is valued, and how success is determined. In contrast, reform is transformative and leads to a redefining of the epistemological position toward the field. Reform raises questions about the core beliefs of mathematics education, moving to restructure thinking about the nature of mathematics, how it is taught, how it is learned, and, ultimately, what constitutes success in learning it.

Throughout the last century, mathematics education in the United States has been a revolving door for revisions—under the guise of so-called reform movements—that failed to question traditional assumptions and beliefs about mathematics teaching and learning and, therefore, failed to change significantly the face of the mathematically successful student. Consequently, speculation exists about whether the current “reform” movement will promote mathematical equity and excellence or turn out to be another trend that fails to significantly alter the status quo in mathematics education (Martin, 2003). We argue that the work currently underway is fundamentally different from past movements. In order to build a case, we will examine the motives and intentions of past revisions in mathematics education— taking care to distinguish them from actual reform. We will then examine current NCTM standards-based efforts to reform mathematics education in contrast to these past movements. A model of shifting paradigms will be offered as a way to understand what makes the current efforts distinct and why they have garnered such sharp criticism from some quarters.

Thorndike’s Stimulus-Response Bond Theory The opening of the twentieth century saw much change in the character of the United States. It was during this time that Edward L. Thorndike, president of the nascent American Psychological Association, led a new class of educational psychologists whose work was aimed at making schools more efficient and effective in educating and stratifying the masses of children who had recently come to populate public schools (Gould, 1996; Henriques, Hollway, Urwin, Venn, & Walkerdine, 1998; Oakes, 1985; Stoskopf, 2002; Thayer, 1928) In particular, Thorndike’s Stimulus-Response Bond theory (Thorndike, 1923) had a profound influence on the teaching and learning of mathematics (English & Halford, 1995; Willoughby, 2000).

Thorndike and his colleagues contended that mathematics is best learned in a drill and practice 8

manner and viewed mathematics as a “hierarchy of mental habits or connections” (Thorndike, 1923, p. 52) that must be carefully sequenced, explicitly taught, and then practiced with much repetition in order for learning to occur. In Thorndike’s work there was an explicit denial of the ability of students to reason about mathematical concepts, as in this example where he explained how students solve 23 + 53: Surely in our schools at present children add the 3 of 23 to the 3 of 53 and the 2 of 23 to the 5 of 53 at the start, in nine cases out of ten because they see the teacher do so and are told to do so. They are protected from adding 3 + 3 + 2 + 5 not by any deduction of any sort but…because they have been taught the habit of adding figures that stand one above the other, or with a + between them; and because they are shown or told what they are to do…In nine cases out of ten they do not even think of the possibility of adding in any other way than the ‘3+3, 1+5’ way, much less do they select that way on account of the facts that 53=50+3 and 23=20+3, that 50+20=70, that 3+3=6…[n]or, I am tempted to add, would most of them by any sort of teaching whatever. (Thorndike, 1923, pp. 68–9)

Thorndike and his colleagues used “scientific” evidence to persuasively argue that mathematics is best learned in a drill and practice manner, leading a large portion of the education community to embrace this view and influencing the teaching of mathematics throughout the twentieth-century (English & Halford, 1995; Glaser, 1984; Willoughby, 2000). Notably, Thorndike’s view of mathematics learning fails to address the nature of mathematical thinking students must apply in problem-solving situations (Wertheimer, 1959). Thorndike’s psychology situates mathematics as a priori knowledge, based on objective reason alone, without taking into account the experiences students bring to mathematics or the meaning they make of what is learned (Brownell, 1935; Resnick & Ford, 1981; Thayer, 1928). This, then, allows students’ mathematics achievement to be discretely measured, quantified, and stratified.

The Progressive Movement The Progressive Education Association (PEA) in the 1920s constructed its movement, in part as a reaction against the highly structured rote schooling practices supported by Thorndike’s theories.

Influenced by John Dewey’s (1899) thinking about schooling and society that emphasized the need to harness and provide direction to the child’s natural impulse toward activities of learning, early progressive educators theorized that learning occurs best when it is The Paradigm Shift in Mathematics Education

connected to students’ experiences and interests (English & Halford, 1995). Among their guiding principles, the PEA asserted that: a) children should have the freedom to develop naturally; b) interest should be the motivation for all work; and c) the teacher is a guide and not a taskmaster (Kliebard, 1987). The early phases of the progressive movement were perceived by many educators to be too radical in their refusal to allow any sort of authority for the teacher and its disregard for organized subject matter.

By the time Dewey (1938) chastised such excesses in Experience and Education the PEA was a relatively minor organization in terms of its impact on schooling practices.

The primary influence of the progressive movement on mathematics education came from a later off-shoot, the social efficiency movement, that emphasized maintaining social order through the differentiation of course placement and instruction.

Although influenced by progressivism’s concern for the learner as an individual, those in the social efficiency movement argued that students “could be guided to expressive self-realization and social integration through scientific educational practices as they were evaluated and trained by experts according to their natural inclinations and abilities” (Holt, 1994).

Social efficiency progressives questioned the importance of mathematics for all students in the secondary curriculum, arguing that for many such coursework was unnecessary (Tyack, Lowe, & Hansot, 1984). They turned toward science in the form of standardized testing which offered proof that certain children were more “able” than others for advanced coursework and, ironically, embraced Thorndike’s work in their development of scientific methods for teaching basic mathematics to the masses of students.

The revisions pressed for by these later progressives proposed that the teaching and learning of mathematics have a utilitarian focus while the study of rigorous mathematical subjects was appropriate for a small elite (see, for example, Bonser, 1932). Studies were undertaken to determine what mathematics content would be of most utility to students outside of school and which students were most likely to succeed in higher-level coursework (Kilpatrick, 1992). The general belief behind these efforts was that the study of advanced mathematics was best suited for those who had a perceived future need for the subject matter— primarily white, middle class males (Willoughby, 2000). Columbia professor William H. Kilpatrick, a former student of Dewey, summed up this sentiment well when he said, “We have in the past taught algebra Mark W. Ellis and Robert Q. Berry III

and geometry to too many, not too few” (in Tenenbaum, 1951, p. 109).

The combined effects of Throndike’s structured “science” of teaching mathematics and the social efficiency movement’s sorting of students into mathematics courses suited to their perceived future needs meant that by the 1940s tracking in mathematics had become standard practice, with most students steered into vocational, consumer, and industrial mathematics courses. This is reflected in the sharp decline in the percentage of high school students taking algebra—from 57% in 1905 to about 25% in the late 1940s and early 1950s (Jones & Coxford, 1970)—that accompanied the dramatic rise in high school enrollment during the same period.

New Math By the mid-twentieth century, a new rationale for the study of advanced mathematics was found in national security and people saw a more rigorous mathematics curriculum as a necessity—at least for some. Congress created the National Science Foundation (NSF) in 1950 in order to develop a national strategy for promoting education in the sciences (Jones & Coxford, 1970). Several projects aimed at overhauling mathematics education received funding from the NSF. At first, “a variety of approaches were being taken to improve mathematical education” (Hayden, 1983, p. 3). As examples of the early “pioneers” of this era, Hayden cites the following: Henry Van Engen and Maurice Hartung developed texts guided by principles of the early progressives; Catherine Stern incorporated the use of manipulatives to deepen conceptual understanding of arithmetic; Robert Davis taught algebra to inner city junior high school students; and Max Beberman and colleagues at the University of Illinois designed a high school mathematics program built around discovery learning that required teachers to attend extensive, intensive workshops before and during implementation. These efforts, while invaluable to the next generation of mathematics educators and their approach to reform, failed to achieve widespread influence in their day.

One group that did obtain a national spotlight for its ideas was the College Entrance Examination Board (CEEB). They established Advanced Placement testing in 1955 with calculus being the first exam offered.

Four years later the Commission on Mathematics of the CEEB issued the final draft of a report, written by a group of fourteen mathematicians and mathematics educators (with just one of the early 1950s reformers 9

included), that recommended changes in the content of the college preparatory mathematics curriculum designed to reflect developments in mathematics such as set theory and Boolean algebra—commonly referred to as modern mathematics (Commission on Mathematics, 1959; Hayden, 1983).

With Russia’s launch of Sputnik in 1957, the United States government became impatient with the slow pace of reform efforts and used NSF funding to create the School Mathematics Study Group (SMSG).

Utilizing the CEEB commission’s report as a guide for its task of modernizing course content, the SMSG swiftly produced and distributed textbooks that “reflected the content and viewpoint of modern mathematics much more completely and accurately than they reflected the pedagogical innovations” (Hayden, 1983, p. 5) of the earlier “pioneer” programs.

These textbooks were sent to schools nationwide in a massive dissemination effort that, for the most part, failed miserably, ultimately leading to calls for a return to more familiar mathematical fare—basic skills.

Critics of the SMSG New Math textbooks, who were numerous (see, for example, Ahlfors et al., 1962; Kline, 1973), claimed that the content was too abstract and not related to real world problems, the language used was unknown to most educated adults, and that more harm than good was the result. The early goals of New Math having to do with issues of pedagogy and access were largely forgotten, leading Willoughby (2000) to conclude that during the New Math era “this apparent recognition of the need for better mathematics education tended to be applied specifically to males of European extraction” (p. 3). Tate (2000) adds to this the insight that most of the federally funded programs now identified as New Math or modern math were developed by mathematicians who believed higherlevel mathematics should be limited to college capable students.

Back-to-Basics In the early 1970s, the “back-to-basics” call was sounded in response to the perceived shortcomings of New Math (Burrill, 2001; National Institute of Education, 1975). This movement called for decontextualized and compartmentalized skillsoriented mathematics instruction and was closely connected to the minimum competency testing movement used extensively by states in the 1970s and 1980s (Resnick, 1980; Tate, 2000). This basic skills mentality dominated textbook publishing through the early 1980s, leading to another generation of Thorndike-like mathematics textbooks (English & 10

Halford, 1995). Although the emphasis on skills did result in slightly improved standardized test scores for students traditionally underserved by school mathematics, it was criticized for not adequately preparing these students for mathematics coursework requiring higher levels of cognition and understanding (United States Congress Office of Technology Assessment, 1992; Tate, 2000).

Failed Revisions Despite a century of “reform” efforts, school mathematics practices in the late twentieth century remained stubbornly similar to what Florian Cajori (1974/1890) described one hundred years earlier in his study of mathematics classrooms across the United States: “[There were] no explanations of processes either by master or pupil…the problems were solved, the answers obtained, the solutions copied” (p. 10).

Research in mathematics classrooms during the latter years of the twentieth century found that in the United States teachers were still the center of authority, first disseminating rote skills and procedural knowledge to their students who then worked individually on sets of problems in order to internalize this knowledge (Cobb, Wood, Yackel, & McNeal, 1992; Fey, 1979; Price & Ball, 1997; Stodolsky, 1988). Likewise, the century of “reforms” did not significantly alter historical assessment methods and patterns of learning outcomes: timed measures consisting of primarily procedural problems showed disproportionate levels of mathematics achievement and attainment across groups of students (Martin, 2003; Schoenfeld, 2002; Gonzales et al., 2004; United States Congress Office of Technology Assessment, 1992).

The behaviorist science of Thorndike’s psychological models, the vocational focus of the social efficiency progressives, the curricular elitism of the New Math program, and the shallow content of the back-to-basics movement have all been referred to as efforts to reform mathematics education but, for the most part, have resulted in superficial revisions to standard practices and outcomes. None of these promoted true reform in mathematics education because they were trapped within an inherently inequitable system of thought toward mathematical knowledge and the teaching and learning of mathematics. The revisions of the past century situated many learners in an a priori deficit position relative to disembodied mathematical knowledge—meaning learning mathematics was taken to be harder for certain groups of students due to their backgrounds and/or innate abilities—and failed to acknowledge the The Paradigm Shift in Mathematics Education

importance of mathematics for all students. Excellence, as defined by these models, meant either remembering rules and procedures with little concern for the connection of mathematics to students’ lived experiences or, in the case of the progressives, focusing on the child’s perceived interests or needs to the exclusion of being concerned with the learning of critical mathematical concepts. In the end, these revisions failed to substantively challenge educators’ thinking about equity and excellence in mathematics education (Martin, 2003; Rousseau & Tate, 2003).

Shifting Paradigms in Mathematics Education The traditional models for mathematics education found within the revisions of the past century have been formulated within a perspective we are calling the procedural-formalist paradigm (PFP). The PFP holds that mathematics is an objective set of logically organized facts, skills, and procedures that have been optimized over centuries. This body of knowledge exists apart from human experience, thus making it inherently difficult to learn.

Thinking within the Traditional Paradigm Positioning oneself within the frame of the PFP, one might reasonably believe the goal of school mathematics education should be for students to internalize a fundamental body of basic mathematical knowledge. In order to facilitate such learning, teachers must deliver carefully sequenced bits of mathematics to students through explanation and demonstration.

Students repetitively practice these facts, skills, and procedures in an effort to memorize them and are then tested to discern what has been “learned.” Learning and assessment are structured around the notion that there is a unique, mathematically correct way to solve a problem. This set of assumptions guided the work of Thorndike and the back-to-basics advocates.

Alternatively, still operating within the PFP, one might try to show students forthright the logical structure of mathematics and hope they catch on.

While many may not, they might at least catch a glimpse of the inherent beauty of mathematics. The students who do catch on to this structure will be well positioned to succeed in higher-level mathematics.

This was the position taken by many of the New Math advocates.

The later social efficiency progressives espoused yet another approach to school mathematics, but one still grounded in the PFP. Since most mathematics is outside of human experience, it is not relevant for students to learn. Any necessary practical mathematical Mark W. Ellis and Robert Q. Berry III

skills can be learned within the context in which they might be used. This thinking led to the creation of consumer and vocational mathematics courses that offered the average student an “escape” from the rigors of such formal topics as algebra and trigonometry (reserving such classes for the mathematically elite).

Importantly, the claim being posited here is not that there were no other possibilities being offered for reform during the past century. Rather, it is that the paths taken—those “reforms” that received strong support and were widely implemented (and the ways in which they were implemented)—were reflective of the PFP. Ideas that fell outside the bounds of this paradigm, though having localized effects, failed to gain wide acceptance, recognition, and/or support.

However, that such efforts were not insignificant is acknowledged for these small-scale deviations from the paradigmatic ways of operating demonstrated to those involved that there was a fundamentally different way to conceptualize mathematics education. In other words, new possibilities were envisioned and experienced, providing the seeds for later efforts at true reform. It would be profitable for further analyses to be brought to the specifics of the ways in which these “fringe” programs had rather important effects on the field.

The Makings of a Paradigm Shift Several strands of thought came together in the 1980s, ultimately leading to what we are characterizing as (the start of) a paradigm shift in mathematics education. Beginning in the mid– to late–1980s several documents were published expressing a concern that, particularly given the trend toward a more technological world, the poor performance of American students in mathematics needed to be addressed in a different way (Leitzel, 1991; National Commission on Excellence in Education, 1983; National Research Council, 1989; Romberg, 1989). We know from examining the past century of mathematics education that such a concern was not new. However, this time there was a co-incidence with the renewed public interest in mathematics education. A new generation of mathematics educators—many of whom came out of K-12 teaching and had been touched by the early professional development efforts of New Math such as Robert Davis’ Big Cities Project (Wilson, 2003)—were looking to a wider body of research to inform their thinking about how best to improve the quality of mathematics learning for all students.

Examining not only the content of school mathematics, these educators also focused attention upon the 11

question of how students learn mathematics. Ideas from mathematical theory were married with theories of learning from cognitive psychology, initially those of Piaget (Rosskopf, Steffe, & Tabeck, 1971) and Bruner (1971), directing attention to the learner’s active role in developing mathematical knowledge.

And, most importantly, the idea that students from diverse backgrounds should all come to understand important mathematical concepts was explicitly endorsed (MSEB, 1990; NCTM, 1989 & 1991; NRC, 1989).

The coalescence of these strands of thought precipitated the release of the National Council of Teachers of Mathematics groundbreaking Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989). While some involved in the authoring of the Standards described them as “more evolutionary than revolutionary” (McLeod, Stake, Schappelle, Mellissinos, & Gierl, 1996, p. 46), the vision they offered re-directed the profession and generated new ways to conceptualize teaching and learning.

Advocates of the Standards realized that what had been missing from American mathematics education was a focus on how students come to form meaningful understandings of and connections between mathematical concepts (Gelman, 1994; Fuson, 1988; Schoenfeld, 1988; Steffe & Cobb, 1988). Students’ engagement with mathematical thinking was given primacy over rote procedural manipulations. Learning came to mean more than memorization and repetition.

Some of the new goals of mathematics education included: making sense of conceptual connections; articulating original insights; explaining and justifying mathematical arguments; and applying knowledge to new situations (Carpenter & Lerher, 1999).

The publication of the NCTM standards was accompanied by a growing awareness of and interest in research examining the significance of culture, specifically the interaction of student culture and classroom culture, to the construction of meaningful mathematical understanding (Cobb, Wood, & Yackel, 1990; Ladson-Billings, 1997; Lave, 1988; Malloy & Malloy, 1998; Saxe, 1991). There was broad recognition that teaching mathematics to all students required something other than the transmission of objective, disembodied content (see, for example, Boaler, 2000). Rather, the ways in which students experienced mathematical ideas and concepts and how this was connected to their own lived experiences came to be seen as critical to the learning process.

Brought together, ideas about the importance of cognition and culture to learning have led to the 12

development of standards, curriculum materials, and pedagogical strategies intended to promote the use of thoughtfully structured experiences and interactions that afford all students opportunities to develop an understanding of the relevance and logic of mathematics.

A New Paradigm Emerges It is the integration of new thinking about cognition and the greater acknowledgement of culture that has enabled mathematics educators to frame questions and conceptualize solutions in ways that were unlikely to develop from within the proceduralformalist paradigm. The unique perspective toward mathematics education that has come from the blending of cognitive psychological and (socio)cultural research has been made possible by the emergence of what we are calling the cognitive-cultural paradigm (CCP). The CCP takes mathematics to be a set of logically organized and interconnected concepts that come out of human experience, thought, and interaction—and that are, therefore, accessible to all students if learned in a cognitively connected and culturally relevant way.

The fundamentals of the cognitive-cultural paradigm lead to a radically different view of mathematics education than that of the proceduralformalist. Many of the core beliefs of traditional paradigm are challenged. Emphasis is shifted from seeing mathematics as apart from human experience to mathematics as a part of human experience and interaction. This is not to imply that students must reinvent mathematics in order to learn it. Rather, for students to really understand mathematics they need opportunities to both a) share common experiences with and around mathematics that allow them to meaningfully communicate about and form connections between important mathematical concepts and ideas, and b) engage in critical thinking about the ways in which mathematics may be used to understand relevant aspects of their everyday lives. The challenge is no longer how to get mathematics into students, but instead how to get students into mathematics (Philipp, 2001). This implies a need for flexibility in how teaching is approached and how learning is evaluated, a move away from more static and deterministic models of the PFP. Conflict Caused by the Paradigm Shift The controversy that has erupted in recent years over how best to approach mathematics education (see, for example, Kilpatrick, 1997; Loveless, 2001; Wu, The Paradigm Shift in Mathematics Education

1997) is evidence of the tension caused by a paradigm shift. In his essay, “What are scientific revolutions?”

Thomas Kuhn (1987) explains that a revolutionary change in a paradigm results in a re-conceptualization of the entire set of assumptions and generalizations within a field and necessitates the development of new vocabulary with which to describe the new ideas.

Looking at mathematics education today, while we find tacit agreement that students’ poor performance in mathematics is problematic, the ideas being proposed to address these woes often seem (and often are) incommensurate.

Viewed from the perspective of the traditional PFP, the ideas for reform coming out of the CCP may appear largely irrelevant and wholly misguided. This explains the competing lists of “excellent” curricula and the contradictory views of assessment in mathematics that have appeared in recent years. If mathematics is a disembodied set of facts, skills, and procedures, it makes little sense to ask students to work collaboratively to construct understanding. Where would this knowledge come from? If mathematics is objective, it makes no sense to be concerned with learners’ cultures and lived experiences. If mathematical achievement can be accurately and fairly measured with standardized tests of routinized items, it makes no sense to develop more “subjective” assessments of mathematical understanding. And if mathematics is inherently too difficult for many to master, it makes no sense to try to teach all students rigorous aspects of the discipline. (See Sfard, 2003 for a thorough and thoughtful discussion of these tensions.)

Beyond the Conflict—From Rhetoric to Reality Disputes aside, curricula designed and implemented in ways aligned with the perspective of the cognitive-cultural paradigm have resulted in what has previously been denied—students of diverse backgrounds, including those from historically underserved populations, have had success in learning not only to solve mathematical problems but also to communicate meaningfully about and value mathematical thinking (see, for example, Boaler, 1997; Campbell, 1996; Knapp et al., 1995; Silver & Stein, 1996; Van Haneghan, Pruet, & Bamberger, 2004). This research has demonstrated that students learning mathematics in reform-oriented classrooms outperform peers in traditionally structured classrooms on not only standard measures of mathematical skill but, more importantly, on measures of mathematical application and understanding. What is more, students in reformMark W. Ellis and Robert Q. Berry III

oriented classes report stronger interest in and motivation toward mathematics (Boaler, 1997; Knapp et al., 1995), important predictors for future coursetaking in mathematics. These efforts have the potential to generate true reform by valuing students’ abilities to make sense of mathematics through meaningful learning experiences including the discussion of mathematical ideas, engagement with non-trivial problem solving tasks, and constructive support from teachers who themselves understand mathematics as more than rote facts, skills, and procedures.

The most salient question for those involved with the CCP-oriented reforms in mathematics education is no longer simply, “Do these reforms work?” While this is still an important query, much effort has moved to thinking about how to prepare teachers—the majority of whom have learned to think about mathematics from within the procedural-formalist paradigm—to understand and implement instructional strategies reflective of the CCP. But what might such work entail?

Working Toward Reform with Teachers of Mathematics A large component of reforming mathematics education in the United States requires asking teachers to think differently about mathematics and to strengthen their own conceptual understanding of mathematics (CBMS, 2001), leading many to reconstruct knowledge that had heretofore seemed disembodied and absolute. Recommendations for instructional strategies situated in the cognitive-cultural paradigm, while eschewing prescriptive formulations of practice, generally call for teachers to structure learning environments that allow for mathematical discourse and the connection of mathematical ideas.

This requires a different and more comprehensive knowledge of mathematics, stretching beyond rote facts, skills, and procedures to include important mathematical ideas and the interconnections among these (Ma, 1999; Sfard, 2003). Such a dramatic shift will not be easy and will not be quick; it will take much time and reflective thinking in order to move one’s practice to a model of teaching oriented so differently than one’s prior learning experiences (Lampert, 2001). Teachers need to see themselves as perpetual learners and be given opportunities to reform their own personal understandings of mathematics (Ellis, 2003; Franke, Carpenter, Fennema, Ansell, & Behrend, 1998). These experiences must be supported by mathematics educators who not only understand but are willing to take up the challenge of reflecting on 13

one’s instructional practices and critically examining the sorts of opportunities that are being created for students to develop mathematical understanding.

Equally important (and under-researched), teaching successfully within the CCP requires that teachers develop understandings of multiple cultures— their own as well as those of their students—and how these are situated within both the local communities of schools and the larger society (Gutiérrez, 2002; Ladson-Billings, 1995; Remillard, 2000). Admittedly, we do not yet understand well the implications this has for our work with teachers. In her explication of what it might take to enable teachers of mathematics to move toward a more culturally cognizant view of their work, Gutiérrez (2002) argues that professional development must explicitly acknowledge teachers and students as members of communities that contribute to and are always in the process of remaking mathematics.

That is, teacher practice aligns with the everyday dilemmas that teachers face, the power that they wield, the influence of local contexts, and the relationships between humans. (p. 175)

She explains that this does not indicate a disavowal of the “traditional” mathematics content but, rather, that such material is to be taken out of its “objective” context and placed it into the real lives and communities of teachers and students. This pushes mathematics beyond being a disembodied set of truths (as conceptualized within the PFP) and challenges us all to examine critically the role mathematics education has played in maintaining and justifying social inequities.

Conclusion By critically examining the past century of “reform” movements in mathematics education in the United States, we have found them to have developed within a common perspective toward mathematical knowledge and mathematical learning that led to inherently inequitable practices and outcomes. An argument was made for conceptualizing the current reform efforts as fundamentally different in that they have come about within a perspective that acknowledges mathematics as culturally situated and views learning as tied to processes of cognition and interaction. This on-going paradigm shift holds the potential to change the look and outlook of successful learners of mathematics from what have historically been relatively small numbers of disproportionately “white” and middle class students, whose learning was focused on solving routinized problems by mastering procedural manipulations, to what should be large 14

crowds reflective of the diversity found within the nation’s public schools who learn to flexibly apply mathematical thinking to investigations of meaningful queries.

However, such an outcome is by no means certain or guaranteed. As we confront the challenges related to teacher preparation outlined above, we must retain a healthy degree of skepticism toward the work we do in order to guard against the simplistic promulgation of yet another set of absolute truths against which students are inequitably provided differential access to opportunities to learn. Indeed, Thomas Popkewitz (2004) has recently offered a critique aimed at what he perceives to be a possible extreme to which the ideas of the NCTM standards (and, by extension, ideas generated within the cognitive-cultural paradigm) may be taken, one in which strictly defined developmental models lead teachers to evaluate students’ learning, ostensibly by criteria such as problem solving skills and the ability to participate in a mathematical community but that end up, in actual practice, promoting the (continued) stratification of students by markers of race and class.

It is in order to steer clear of such dangers that many involved with the reforms of the cognitivecultural paradigm deliberately refrain from issuing rigid prescriptions for classroom practices and static lists of criteria for measuring learning, opting instead to share varied descriptions of learning environments and multiple examples of the sorts of outcomes to be expected (see, for example, Stein, Smith, Henningson, & Silver, 1999). Though this approach is at times criticized for its failure to provide concrete directives and quantifiable “objective” indicators, this may be taken as yet one more sign of the shifting paradigms in our midst.

It is clear that the work of reform requires large investments of time and energy in order to enact critical change in mathematics education. What sustains us as we engage in this work is thinking about what the future may hold when students of all backgrounds develop meaningful and powerful understandings of mathematics. While not claiming to know exactly what such a future may mean for society, we are hopeful it will at least be less inequitable than that of today.

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The Paradigm Shift in Mathematics Education

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The Mathematics Educator 2005, Vol. 15, No. 1, 18–24

INTEGRATING TECHNOLOGY INTO THE MATHEMATICS CLASSROOM: THE ROLE OF TEACHER PREPARATION PROGRAMS Regina M. Mistretta Today’s technology standards (International Society for Technology in Education, 2000) challenge teacher education programs across the nation to address the need to produce computer literate teachers who are confident in their ability to choose and incorporate instructional technology into their classroom teaching. The purpose of this paper is to present a description and analysis of the effects of training sessions on pre-service teachers’ ability to evaluate and integrate instructional technology into the mathematics classroom. A group of 70 pre-service teachers participated in training sessions where they evaluated the features of various Pre K-12 mathematics instructional software and websites, and designed technology-based mathematics lessons. The training sessions promoted the pre-service teachers’ awareness of, appreciation for, and confidence in their ability to analyze, select, and craft technology-based mathematics lessons.

Today’s technology standards (ISTE, 2000) challenge teacher education programs across the nation to address the need to produce computer literate teachers who are confident in their ability to choose and incorporate instructional technology into their classroom teaching. For this reason, it is crucial for teacher educators to share effective ways to prepare pre-service teachers to be able to incorporate technology into their future classrooms. This paper presents a model of how teacher educators can be catalysts in producing teachers who are prepared to integrate technology into the mathematics classroom and suggests directions for future research in this area.

Literature Review Wenglinsky (1998) used data from the 1996 National Assessment of Educational Progress (NAEP) in Mathematics to study the effects of teachers’ use of instructional technology on student achievement in mathematics.

Findings revealed that when appropriately used, computers may serve to improve student mathematics achievement as well as enhance the overall learning environment of the school.

Teachers who received training in the area of instructional technology were found more likely than those who had not to use computers in effective ways such as in simulations, applications, and math learning games. Wenglinsky (2000) also used the 1996 NAEP Dr. Regina Mistretta is an assistant professor in The School of Education of St. John’s University. Her area of interest is Mathematics Education. She has taught at elementary, middle, and secondary levels. Presently she teaches both undergraduate and graduate level courses and is conducting mathematics professional development for schools in the metropolitan area.

Regina Mistretta

data to show the positive effects of using instructional technology to nurture higher order thinking skills in the mathematics classroom.

Unfortunately, Lederman and Neiss (2000) report that technology courses which are part of teacher preparation programs often emphasis pre-service teachers’ learning about technology rather than the integration of technology into classroom teaching. The need for teacher preparation programs to serve as catalysts for the integration of technology into classroom instruction is vital. Abilities, knowledge, and skills in teaching with technology need to be emphasized in the preparation of teachers so that they can make informed decisions about which technology to use for specific teaching purposes.

Linking technology with curriculum has caused significant changes in teaching and learning. Wright (1999) reports higher student achievement, selfconcept, attitude, and teacher-student interaction as a result of interactive learning made possible via technology. Kerrigan (2002) has found the benefits of using mathematics software and websites to include promoting students’ higher-order thinking skills, developing and maintaining their computational skills, introducing them to collection and analysis of data, facilitating their algebraic and geometric thinking, and showing them the role of mathematics in an interdisciplinary setting. As a result of such research, Neiss (2001) reports the National Council of Teachers of Mathematics pinpoints technology as an essential component of the Pre K-12 mathematics learning environment, influencing the mathematics that is taught as well as enhancing students’ learning.

Despite these results and growing access to technology, Kent (2001) reports the U.S. Department 18

of Education estimates that only 20% of all public school teachers feel comfortable using technology in the classroom. Of these teachers, 99% have access to computers and the internet somewhere in their schools.

However, only 39% reported frequent use of computers or the internet to create instructional materials; 34% used them for record-keeping; and less than 10% used them to access lesson plans, do research, or investigate best practices.

Today’s technology standards (ISTE, 2000) challenge teacher education programs across the nation to address the need to produce computer literate teachers who are not just knowledgeable of the internet, word processing programs, spreadsheets, and presentation software, but are also confident in their ability to incorporate instructional software and websites into everyday classroom teaching. Cesarone (2000) reports the National Council for Accreditation of Teacher Education Task Force has recommended more effective uses of technology in teacher education programs, and Halpin (1999) urges teacher educators to determine effective ways to prepare pre-service teachers to integrate technology appropriately into classroom instruction.

Teachers’ ability to select appropriate software and websites is an essential component of the ultimate success of effectively integrating instructional technologies into classroom teaching. Ertmer, Addison, Lane, Ross, and Woods (1999) state that “teachers, not technology, hold the key to achieving integrated technology use,” while Haughland (2000) states that how computers are used is more important than if computers are used.

The research literature causes one to reflect on the powerful influence that instructional technology might have on teaching and learning if utilized properly.

Although research has suggested that the use of technology can improve student achievement and self efficacy, many reports demonstrate that teachers use computers minimally and many are unprepared to integrate technology into their classrooms. One way to prepare teachers to integrate technology into their classrooms is for teacher educators to work with teachers to improve their understanding of, and ability to utilize, technology in meaningful ways in the classroom. This paper describes work done with preservice teachers that seeks to address precisely how it is that teacher educators can work towards addressing this need.

A group of 70 pre-service teachers participated in training sessions in which they evaluated the features of various PreK-12 mathematics instructional software Regina M. Mistretta

and websites and designed technology-based mathematics lessons tailored to the content they anticipated teaching. The training sessions were integrated into a required course of the teacher education program that focused on technology in the school, community, and workplace. The pre-service teachers were undergraduate students in their second year of study and were registered in one of four course sections taught by the same professor.

The sessions attempted to broaden pre-service teachers’ knowledge about, and strengthen their ability to select, software and websites for specific uses in the mathematics classroom. Through active participation in collaborative activities involving the development of technology-based lessons, the training sessions also sought to improve the pre-service teachers’ confidence levels as practitioners of technology-based instruction.

Descriptions of the training sessions as well as the preservice teachers’ reactions to them are shared to broaden the knowledge base and serve as a step towards specifically guiding the ways teacher educators prepare their pre-service and in-service teachers for today’s technology-based mathematics classrooms.

Initial Assessment An initial assessment was conducted to guide the design of the training sessions. The pre-service teachers completed a questionnaire about their previous experience with instructional technology and their self-confidence in reviewing and integrating software and websites into the teaching of mathematics. They were asked to supply the specific software titles and websites they had previously used at school, home, and/or work, along with an explanation of how and for what purposes the software and websites were used. Interviews with all of the preservice teachers were also conducted. These interviews allowed participants to speak candidly about their written responses and allowed the interviewer to form a deeper understanding of the pre-service teachers’ comments.

The initial assessment indicated that 91% of the pre-service teachers had used Word Perfect or Word for typing research papers, 43% had designed presentations with PowerPoint, 33% had created spreadsheets with Excel, and 96% of the pre-service teachers had used the internet mainly for the purpose of e-mail. The assessment also showed only 30% of the pre-service teachers had used software for instructional purposes. Of this 30%, 9% used it at home for standardized test preparation. 4% of the pre-service 19

teachers had used reading and math readiness software with children, ages 4-6, in an after-school program.

17% of the pre-service teachers used software when they were students in school, but only 3% had used it regularly in either science or math classes. The remaining ten reported sporadic use of software that did not relate to classroom topics.

All of the pre-service teachers felt insecure about their ability to review mathematics software and websites. Their comments revealed they “would not know where to begin” if asked to select instructional technology or to design a technology-based lesson for the mathematics classroom. A common misperception was software and websites used in classrooms contained only games that students could play once content was mastered.

Training Sessions

integration of software and websites into the teaching of mathematics. The training sessions allowed the instructor to prepare the pre-service teachers to enter their future mathematics classrooms not only knowledgeable about the capabilities of instructional technology, but also experienced enough to confidently review and appropriately integrate their selected software and websites into classroom teaching. The training sessions required the pre-service teachers to collaborate and actively engage in using the skills they were learning. The rationale for such a feature stemmed from human development theorist Lev Vygotsky who emphasized “the importance of social relations in all forms of complex mental activity” (p.10), and other constructivist advocates who stress that teachers can most effectively acquire new knowledge through collaboration with other teachers and teacher educators (Prawat, 1993).

The pre-service teachers participated in training sessions (see Table 1) involving the review and Table 1 Training Sessions Training Session Type Introductory Sessions

Guided Evaluation Sessions

Paired-Learning Sessions

Lesson Planning Sessions

Description Provided pre-service teachers with demonstrations of instructional technology leading to discussions concerning its use in the mathematics classroom and development of a set of evaluation criteria.

Provided pre-service teachers with the opportunity to evaluate instructional technology and acquire insights from one another by sharing findings and opinions.

Provided pre-service teachers with the opportunity to collaborate and actively participate in the instructional technology evaluation process.

Provided pre-service teachers with the opportunity to use newly acquired skills to appropriately select and integrate technology into classroom mathematics lessons.

Introductory Sessions The training sessions began with three 1-hour whole group instructional class periods that involved the professor demonstrating a piece of mathematics software and a mathematics website for the elementary (Pre-K-5), middle (6-8), and secondary (9-12) school levels. The professor guided discussion and posed questions so that the pre-service teachers could share their thoughts about the characteristics of the instructional technology, determine the purpose

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Length Three 1-hour sessions

Three 1-hour sessions

Three 1-hour sessions

Two 1-hour sessions

(conceptual development, skill building, promotion of mathematical reasoning/higher order thinking skills) of the software and websites, establish the type of instruction possible (individual, paired, small group, large group, whole group), and assess the benefits of using the software and websites as opposed to other teaching tools. This exchange of ideas provided the framework from which evaluation criteria were developed (see Table 2).

Integrating Technology

Table 2 Evaluation Criteria adapted from Roblyer, 2003 1.

2.

3.

4.

5.

Connection to mathematics standards.

Appropriate approach to mathematics topics with respect to grade, ability, and reading level(s).

Worthwhile mathematical tasks.

Presence of conceptual development, skill building, and problem solving/higher order thinking skills.

Use of practical applications and interdisciplinary connections.

Guided Evaluation Sessions During the next three 1-hour training sessions, the pre-service teachers used the evaluation criteria to review a piece of mathematics software and a mathematics website for each of the grade levels: elementary, middle, and secondary. As the students and professor together assessed the instructional technology, each of the evaluation criteria was addressed. The similarities and differences among the software and websites were addressed as well as the specific use of each software and website in the classroom. The predominate strengths were noted and used to determine those programs and websites best suited for specific classroom settings and objectives such as whole group instruction, small group instruction, conceptual learning, skill building, problem solving, practical applications, and interdisciplinary connections. For example, the preservice teachers distinguished how the mathematics software and websites that contained manipulatives, visual displays, and multiple representations would be appropriate for conceptual development, while those that engaged the students in real-life situations such as owning a business and designing architectural blueprints were best suited for problem solving and practical applications.

The pre-service teachers also considered the interactive features by which students were given feedback about their responses and were guided to explore multiple methods of solution. They looked for assessment and record-keeping aspects of the software and websites that would aid in the evaluation of students’ conceptual understanding, computational skill, and problem solving ability. The pre-service teachers also noted instructional technology that promoted writing in the mathematics learning process.

Throughout the discussions, one of the pre-service teachers volunteered to record and report the sessions to the whole group. These notes were used to provide a

Regina M. Mistretta

summary of the discussions and as a way to promote reflection on the pre-service teachers’ findings.

Paired- Learning Sessions For the next three 1-hour training sessions, the preservice teachers were paired and asked to review an additional 6 items of instructional technology that included a piece of software and a website for each of the elementary, middle, and secondary levels. They were instructed to use the previously developed evaluation criteria to guide their reviews. Throughout this assignment, the students were asked to note their level of engagement and their degree of comfort with the evaluation process. Conferences between each of the pairs and the professor were held to discuss the instructional purpose and appropriateness of the software and websites they were evaluating.

When all pairs completed their evaluations, the professor had them discuss their findings and recommendations with the entire group. The preservice teachers justified their conclusions by demonstrating specific features of the software and websites to their classmates. Feedback from the entire group was elicited after each of the evaluations, giving everyone the opportunity to share ideas.

Lesson Planning Sessions The same student pairs were given the task of designing a technology-based mathematics lesson plan for an assigned topic and grade level. During a 1-hour training session, the professor assigned the topics and grade levels and explained the requirements of the task.

These requirements included the typical structure used for planning a lesson (aim, behavioral objective, motivation, materials, procedure, summary, evaluation, follow-up) used in all of the education courses of the pre-service teachers’ education program. They were to select and integrate either a piece of software or a website as their main technological teaching tool. The lesson plan follow-up was to involve a piece of instructional technology that was not used in the main portion of the lesson but would extend the lesson’s objectives. For example, if a team developed a lesson about multiplying fractions for grade four using a piece of software, their follow-up could use a website featuring multiplication of fractions for grade four. The professor shared a checklist (see Table 3) with the preservice teachers to guide the development of their technology-based lesson plans.

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Table 3 Checklist for Development of Technology-Based Lessons adapted from Roblyer, 2003 1

Should activities be individual, paired, small group, large group, whole class, or a combination of these?

2

What instructional activities need to come before the introduction of the technology resource?

3

What instructional activities need to follow the introduction of the technology resource?

4

How will you assess students’ learning progress and products?

5

Should the lesson take place in a lab or will classroom computers be adequate?

6

Will you need projection devices or large screen monitors?

to help them take notes and critique the lessons. The presenters elicited feedback from the class and responded to their classmates’ comments, questions, and suggestions. Whole group discussions followed each presentation and provided a forum for the exchange of ideas and constructive criticism.

Each of the pre-service teachers critiqued their own lesson in written form as a way to reflect on their lessons as well as their classmates’ comments and suggestions. Within these written reflections, the preservice teachers discussed the strengths and weaknesses of their lessons along with ways to improve them. These self-assessments were a culminating element of the training sessions that portrayed the pre-service teachers not only as active participants in the evaluation process, but also as reflective practitioners of technology-based mathematics instruction.

Reactions

During another one-hour training session, the preservice teachers shared what they envisioned as the characteristics of a technology-based lesson. A set of lesson plan criteria (see Table 4) emerged from this dialogue and provided a framework to assess the quality of the lessons. Students received guidance and feedback throughout the remainder of the semester from both the professor and fellow classmates on lesson plans designed to incorporate instructional technology.

Table 4 Lesson Plan Criteria developed by Roblyer, 2003 1

The technology activity is a seamless part of the lesson.

2

The students are focusing on learning with the technology, not on the technology itself.

3

Lesson objectives could not be accomplished or accomplished as well if the technology weren’t there.

4

The contributions of the instructional technology are evident.

5

All students are engaged with the technology and benefiting from it.

Presentations Each pair of pre-service teachers presented their lesson plans using PowerPoint and engaged the class in using their selected piece of software and website. The pre-service teachers who were not presenting the lesson used the lesson plan criteria developed in class 22

Upon completion of the training sessions, the preservice teachers were given a second questionnaire.

They were asked to reflect on their experiences reviewing and integrating mathematics instructional technology, their understanding of the uses of technology in the mathematics classroom, and their comfort level teaching mathematics using technology.

The pre-service teachers were specifically asked to comment on the positive and negative aspects of the training sessions, the most beneficial uses of mathematics instructional software and websites, their self-confidence in evaluating and integrating instructional technology into mathematics lessons, and their attitudes towards themselves as mathematics educators using technology-based instruction.

Interviews were again conducted with all of the participants so that they could speak candidly about their written responses. Patterns in responses revealed the pre-service teachers’ positive evaluation of the training sessions, increased awareness of the power of instructional technology, and heightened levels of confidence in their ability to use technology-based mathematics instruction.

The questionnaires revealed that 94% of the preservice teachers viewed the training sessions as a completely positive experience. Of those responding that the training sessions were not completely positive, the only negative aspect reported involved technical difficulties that infrequently occurred when a program wouldn’t run or a website wouldn’t respond.

Responses showed that the pre-service teachers valued using a set of evaluation criteria when reviewing Integrating Technology

software and websites, they enjoyed working with a partner as well as sharing their work with others in the class, and they indicated that the “support system” present within the design of the training sessions guided their inquiries and gave them opportunities to brainstorm with others to refine their ideas.

Eighty-six percent of the pre-service teachers indicated that they viewed software and websites as “great tools” that can promote best practices and provide content information and motivational activities. Their comments included: “Instructional technology can enhance the mathematics learning environment by providing visual demonstrations, interdisciplinary connections, and practical applications,” “Teachers don’t have to hunt for information about math topics,” “Software and websites can readily give teachers the information they need to understand the topics they teach,” and “Instructional technology helps the teacher bring mathematics to life with real-world connections.”

Fifty-seven percent of the pre-service teachers responded that they were more aware of the tremendous potential technology had for helping students to conceptualize and apply mathematics. A common response was “Instructional technology allows teachers to plan and implement mathematics lessons with ease and efficiency that would otherwise take them months to prepare and facilitate.” They marveled at the capability of software and websites to model concepts using multiple representations, integrate writing into the mathematics classroom using electronic journals, and link mathematical concepts such as ratio and proportion and geometric relationships to architectural blueprints and artistic designs. They also looked forward to involving their future students in using instructional technology to run a mock business while learning about practical applications of mathematics.

Eighty-three percent of the pre-service teachers commented that they were more confident in the process of evaluating software and websites. Their responses indicated greater self-confidence in appropriately selecting and integrating instructional technology. Their responses included: “We didn’t just accept the first thing we saw,” “We selected tools that were relevant, motivational, and would really benefit kids,” and “We were able to focus on the software and website features and see both the pros and the cons of their use in the classroom.” The pre-service teachers commented that the development of technology-based lessons was an experience that “put theory into practice.” Integrating technology into their own lessons Regina M. Mistretta

seemed to “open a door to a different world of teaching and learning” for them. Their comments included statements such as: “It was helpful to have us present our lessons and critique each other as well as ourselves,” “We seemed to learn from each other rather than from what we were just assigned,” and “Actually incorporating a piece of software and a website into my own lesson gave me the opportunity to implement what I learned to do.”

Conclusions and Recommendations Research reveals positive effects on teaching and learning when technology is used to its fullest potential. It is therefore important that teacher education programs determine effective ways to prepare teachers to integrate technology into their classrooms. The study described in this paper investigated the effects of engaging pre-service teachers in training sessions designed to prepare them to integrate technology into mathematics instruction.

Findings revealed that the training sessions promoted the pre-service teachers’ awareness of, appreciation for, and confidence in their ability to analyze, select, and craft technology-based mathematics lessons. This study serves as a stepping-stone for future research.

Longitudinal investigations involving comparisons of confidence levels and lesson quality among classroom teachers who participate in the instructional training described in this paper with those who don’t would further define the effects of the training. Incorporating and studying the effects of the training sessions in professional development programs and graduate courses would also provide deeper insight into its influence on in-service teachers as well as pre-service teachers.

Training teachers to integrate technology coupled with continued investigation into its effects on teaching and learning serve to empower technology-based learning environments. This study’s model proved to be successful and can be used by teacher educators in technology courses as well as mathematics methods courses. The research efforts presented in this paper exemplify how higher education can serve as a catalyst towards effective use of instructional technology.

REFERENCES Cesarone, B. (2000). Teacher preparation for the 21st century.

Childhood Education, 76(5), 336–338.

Ertmer, R., Addison, P., Lane, M., Ross, E., & Woods, D. (1999).

Examining teachers’ beliefs about the role of technology in the elementary classroom. Journal of Research on Computing in Education, 32(1), 54–66.

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Halpin, R. (1999). A model of constructivist learning in practice: Computer literacy integrated into elementary mathematics and science teacher education. Journal of Research on Computing in Education, 32(1), 128–135.

Haughland, S. W. (2000). What role should technology play in young children’s learning? Part II: Early childhood classrooms in the 21st century: Using computers to maximize learning.

Young Children, 55(1), 12–18.

International Society for Technology in Education. (2000).

National educational technology standards for students: Connecting curriculum and technology. Eugene, OR: Author.

Neiss, M. (2001). A model for integrating technology in pre-service science and mathematics content-specific teacher preparation.

School Science and Mathematics, 101(2), 102–109.

Prawat, R. (1993). The value of ideas: Problems versus possibilities in learning. Educational Researcher, 24(7), 5–12.

Roblyer, M.D. (2003). Integrating educational technology into teaching. Upper Saddle River, NJ: Pearson Education, Inc. Wenglinsky, H. (1998). Does it compute? The relationship between educational technology and student achievement in mathematics. Princeton, NJ: Educational Testing Service.

Kent, K. (2001). Are teachers using computers for instruction? The Journal of School Health, 71(2), 83–84.

Wenglinsky, H. (2000). How teaching matters: Bringing the classroom back into discussions of teacher quality. Princeton, NJ: Educational Testing Service.

Kerrigan, J. (2002). Powerful software to enhance the elementary school mathematics program. Teaching Children Mathematics, 8(6), 364–377.

Wright, R.T. (1999). Technology education: Essential for a balanced education. NASSP Bulletin, 83(60), 16–22.

Lederman, N., & Neiss, L. (2000). Technology for technology’s sake or for the improvement of teaching and learning? School Science and Mathematics, 100(7), 345–348.

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Integrating Technology

The Mathematics Educator 2005, Vol. 15, No. 1, 25–34

Predicting Turkish Ninth Grade Students’ Algebra Performance Ayhan Kursat Erbas The prediction of students’ achievement in algebra in eighth and ninth grades has become a research interest for practical issues of placement. A group of simple, easily accessible variables was used to predict student performance in algebra after completion of eighth grade. The three variables of school type, grade level, and previous year mathematics performance explained 54% of the variance in algebra performance. Furthermore, school type was the dominant predictor of performance, explaining 33% of the total variation in algebra achievement.

Accepted as a gatekeeper, algebra has been a major focus of school mathematics. The basic reason may be the power that algebra has provided for operating with concepts at abstract levels and applying those concepts in concrete situations, beginning with the invention of “symbolic algebra” by Vieta (Kieran, 1992). The place of algebra in school curricula differs around the world (Howson, 1991), as some countries offer integrated or unified mathematics in their schools and others approach mathematics as a unified body of strands such as algebra, arithmetic, and geometry.

In the United States, a differentiated high school mathematics curriculum (i.e., first-year algebra, geometry, and second-year algebra followed by trigonometry and/or precalculus, and calculus), taught in a linear fashion, appears to be the standard format, differing from state to state in terms of mathematical content and scope. Traditionally, algebra has been a part of the college preparatory mathematics curriculum, offered to only a fraction of students because of tracking or choices made by students, teachers, and parents. For many years, it was offered as a single course or two, promoting manipulative skills with which students were expected to master manual techniques for transforming, simplifying, and solving symbolic expressions that were often isolated from a real-life context. However, the reform movement promoted by the National Council of Teachers of Mathematics (NCTM, 1989, 2000) suggests that school mathematics in general, and algebra in particular, be considered an integrated and cohesive body of concepts, each closely related to the other major strands of school mathematics. With the reform Ayhan Kursat Erbas is an Assistant Professor in the Department of Secondary Science and Mathematics Education at the Middle East Technical University, Ankara, Turkey. His research interests include teaching and learning of algebra, integrating technology into mathematics education, and teacher knowledge and beliefs.

His e-mail is erbas@metu.edu.tr Ayhan Kursat Erbas

movement, the distinction between pre-algebra, algebra, and other courses offered as a part of a general mathematics curriculum has become less apparent. The development of an informal understanding of algebraic ideas in grades K-8 is now not only acknowledged but also encouraged, with the hope that all students will have a strong foundation in algebra and geometry by the end of eighth grade and a strong desire to pursue higher algebra and mathematics in high school. In many school districts, Algebra I courses, traditionally offered to ninth grade students and a selective few eighth graders, are now available to all students in eighth grade.

The Turkish educational system endorses a vision of integrated mathematics and a national school mathematics curriculum in which all students are offered the same compulsory mathematics courses until the end of eighth grade. It is more or less the same in secondary education (ninth, tenth, and eleventh grades); and mathematics, and thus algebra, is again compulsory for all students. However, the scope of mathematical concepts may differ somewhat for those students who decide to take the university entrance exam with a social sciences emphasis. This decision is made at the beginning of tenth grade, and those students are required to take fewer mathematics courses than the students whose decisions favor an emphasis on science and mathematics. The situation is similar to that of students in the United States who choose to attend magnet schools.

Formal algebraic concepts are presented to all students and generally begin with an introduction of literal expressions and linear equations in seventh grade and continue with more emphasis in the eighth grade. More specifically, the seventh and eighth grade mathematics curricula prepared by the Committee of Teaching and Training in the Turkish Ministry of National Education and offered to all students in those grades have the following aims: 25

Seventh Grade: • understanding Mathematical Expressions, • knowledge of Proposition, Open Proposition, and Equations, • ability to solve linear equations with one unknown, • ability to solve linear inequalities with one unknown, • understanding the coordinates of a point in the Cartesian plane, • ability to draw graphics.

Eighth Grade: • ability to make operations with literal expressions, • understanding Binomial Expansion, • understanding certain identities, • ability to factorize, • ability to solve linear equations with one unknown, • ability to solve linear equations with two unknowns, • understanding the equation of a line, • understanding linear inequalities with two unknowns, • ability to solve linear inequalities with two unknowns.

• understanding basic trigonometric identities.

The following summary of content of mathematics courses in ninth, tenth, and eleventh grades may give an idea about the place of algebra in the secondary mathematics curriculum: •

•

•

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The ninth grade mathematics curriculum consists of the following concepts and topics: logic; sets; relations, functions, and operations; numbers (natural numbers, integers, modular arithmetic, rational numbers, real numbers, absolute value, exponential numbers, and radical expressions); polynomials; and quadratic equations, inequalities, and functions.

The tenth grade mathematics curriculum includes the following concepts and topics: trigonometry; complex numbers; logarithms; permutations, combinatorics, and probability; induction; sequences and series.

The eleventh grade mathematics curriculum consists of the following concepts and topics: functions and advanced graphing (including special functions such as sine functions, absolute value functions, step functions, etc.); limit; continuity; derivations; integration; and linear algebra (matrices and determinants).

The teaching and learning of algebra in a Turkish context is not much different from what is seen in an American context. In the traditional mathematics course setting, students are introduced to the main points of algebra topics; given the rules and formulas involving manual skills for transforming, simplifying, and solving symbolic expressions, followed by a few rote examples; and then the students are expected to solve similar exercises and problems. Although they are widely available and used even in traditional algebra classrooms in the United States, manipulatives like algebra tiles are generally not available and are not used in the teaching and learning of algebra in Turkey.

However, other recommendations of the reform movement in mathematics education have become more visible in Turkish schools; and over the last few years, the national K-8 mathematics curriculum has gone through major changes in terms of content and instructional strategies with more student-centered teaching, use of manipulatives, and utilization of technology, particularly calculators. It also requires the earlier introduction of algebraic concepts, starting in K-5 with scale models and the investigation of various number and shape patterns. The new curriculum, with revised teaching/learning tools and materials, is almost ready for implementation, and groups of lead teachers began in-service training in the 2004-2005 school year.

The Ministry of National Education has opened a textbook writing competition among publishers for the new K-8 curriculum; and seven textbooks, from among more than 100, will be selected by a committee and then recommended to all schools nationwide for the 2005-2006 school year. The duration of high school will be extended from three to four years, and research for the development of a new 9-12 mathematics curriculum has begun.

Not only whether algebra is offered to all students but also the ages and grade levels at which algebra is introduced differs from country to country (Howson, 1991), as we can see from the Turkish and American examples. Yet, one thing that might be common everywhere is that the road to algebra is never as smooth as one may wish. Algebra is one of the areas in which students have major problems (Booth, 1988; Kieran, 1992), and identifying student difficulties and measuring student achievement have gained importance as a focus of various research studies (Booth, 1984, 1988; English & Halford, 1995; Flexer, 1984; Herscovics, 1989; Kieran, 1989, 1992).

The prediction of students’ achievement in algebra in eighth and ninth grades has become a research interest for practical issues of placement (Flexer, Predicting Algebra Performance

1984). Students gender, intelligence (IQ), socioeconomic status, previous mathematics performance, and standardized test scores have often been used as predictors of achievement in algebra or mathematics (Flexer, 1984). However, the usefulness of these predictors is limited because not every country has an educational system that includes the use of standard achievement or intelligence tests. Thus, it is necessary to be able to predict student algebra achievement using an accessible and small number of predictors. The purpose of this study was to predict, at the beginning of ninth grade, student performance in algebra, using the variables of school type, previous year mathematics performance, grade level, and gender.

Method Sample The participants in the study were 217 students from two public academic, one private, and one vocational-technical high school in a socioeconomically middle-class district of a metropolis in Turkey. This school district was selected after a pilot study had been carried out in the district to develop some of the test instruments for this study, and the schools were selected by a random sampling method (Fraenkel & Wallen, 1996) according to their type: private school (PrSc), public academic school (PuAc) and vocational-technical high school (VoTe)1. Two public academic schools were chosen because the number of such schools was greater than the number of the other two types. Two classes from each school, eight in all, were chosen based on availability and permission of the participant schools. According to Green (1991), for a power of .80 (alpha = .05), four predictors and a medium effect size, the desirable sample size is around 85. The sample size in this study was large enough to conduct a regression analysis.

Predictor Variables Four predictor variables were considered in the present study: (i) school, (ii) previous year mathematics performance, (iii) gender, and (iv) grade level. Data for the predictor variables were measured by a short questionnaire administered along with an algebra test used to measure the criterion variable, performance. The collection of predictor variables, along with a description of the method by which they were measured, is given in Table I.

Ayhan Kursat Erbas

Table I Predictor Variables and their Measurements for the Regression Analysis Predictor Variable

Measurement

School

1 = PrSc 2 = VoTe 3 = PuAc-1 4 = PuAc-2

Previous year mathematics performance

1 = Minimum to 5 = Maximum

Gender

0 = Male 1 = Female

Grade level

1 = Preparatory 2 = Ninth Grade

In order to measure the prior mathematic performance, students were asked to provide the grade they had received in their mathematics course at the end of the previous year. In the Turkish educational system, grading is based on a five point scale corresponding to 1 = F, 2 = D, 3 = C, 4 = B, and 5 = A. Thus, as an ordered variable, previous year mathematics performance was scored by integer scaling, ranging from 1 to 5. The scoring for the two dichotomous variables was 0–1 for gender and 1–2 for grade levels as shown in Table I. Criterion Variable To measure student performance, I developed an Algebra Test (AlTe) and used the first version in a pilot study. Subsequently, some items in the test were rearranged and revised, and in its final form, the test consisted of 25 open-ended items covering 7th and 8th grade algebra topics. While developing the test, I took the new mathematics curricula for seventh and eight grades, developed by the Education and Training Board of the Turkish Ministry of National Education (1991), into consideration. The algebra topics included on the test were generalization, factorization, literal terms, transformation, first order linear equations (with one and two unknowns), first order linear inequalities (with two unknowns), and word problems. Three of the items had been used in previous research and reported in the literature: two items used in the Concepts in Secondary Mathematics and Science (CSMS) project

by KĂźchemann (1981) and one item used by Whitman (1976) were used with permission. The alpha reliability of the test (Cronbach, 1951) was .76. Sample items and some information asked in AlTe are included in Appendix B. 27

AlTe was administered during the fall semester by the mathematics teachers of the participating students because the students might not have taken the test seriously if they thought it came from outside the school and would not affect their grades. The teachers were informed about the test and how to administer it before they gave it to the students in their classrooms.

The students were given no time restriction for completion of the test and turned in their answer sheets when they felt that they were finished with everything they wanted to do. On the average, the testing time was a class period, which was 45 minutes.

Results The data analysis was carried out in two steps: preliminary data analysis and multiple regression analysis. The means (M), standard deviations (SD), and correlations among all variables for students in the

study are presented in Table II. The male students, the ninth graders, the students having a grade of 3 in the previous year’s mathematics course, and the students attending a public academic school were dominant profiles in the sample. The students' low performance on the Algebra Test is apparent, and the large standard deviation is an indicator of high variance in the sample.

The correlations among the four predictors ranged from r = -0.489 to r = 0.503. Among those, school type was moderately correlated with the students' previous year mathematics performance and grade level.

Correlations between each predictor and student performance on the algebra test performance ranged from r = -0.549 to r = 0.589. In particular, scores were moderately correlated with school type and students' previous year mathematics performance.

Table II Descriptive Data for and Intercorrelations between Predictor and Dependent Variables Variable

1

2

3

4

School Students` previous year mathematics performance

-.489** (.000)

Gender

.083 (.232)

.097 (.162)

Grade levels

.503** (.000)

-.191** (.006)

.167** (.016)

Algebra Test

-.549** (.000)

.589** (.000)

.033 (.635)

.034 (.625)

M

SD

2.75

1.03

2.89

1.19

.37

.48

1.83

.38

15.49

13.33

** Value is statistically significant with the corresponding p-value shown in italics inside the brackets.

To assess the relative contribution of each predictor variable as to the variation of the prediction of performance, an analysis was carried out by deleting each predictor from the analysis. The results of this process are summarized in Table III. The results show that school type contributed more to the prediction of algebra performance than did any of the other three predictors. For these data, there was not much difference in the relative importance of the students' previous year mathematics performance and their grade levels, and the students' gender contributed the least to explaining the variations in the prediction. In fact, the R-squared value was almost unaffected when gender was removed from the regression equation.

28

Table III Relative Contribution of Predictors to Accuracy of Prediction.

2 Radj

Predictor deleted

Rank

School

0.35

1

Students` previous year mathematics performance

0.42

2.5

Grade levels

0.44

2.5

Gender

0.54

4

Predicting Algebra Performance

Although the number of predictors was small, I conducted an all subset analysis (Pedhazur, 1997, p.

212), with results summarized in Table IV. According to this analysis, school type alone was the best predictor and explained 33% of the variance in algebra performance. School and previous year math performance explained 44% percent of the variation; 54%, when grade level was included.

Table IV All-possible-subsets analyses for regression Subset size

Predictors in best subset

1

School

0.33

2

School, Previous Math Grade

0.44

3

School, Previous Math Grade, Grade Level

0.54

2 Radj

Finding that school was a fairly good predictor of algebra performance, one may wonder which schools

did better. Figure 1 shows a boxplot display of algebra scores by schools. Figure 2 gives the same information by previous grade in mathematics.

According to Figure 1, students in the private school (PrSc) did relatively better than their counterparts in the public academic schools (PrSc-1 & PuAc-2) and the vocational technical school (VoTe). Although the students in PuAc-1seemed as competent as the students in VoTe, the performance of students in PuAc-2 seemed very poor compared to the other three schools.

Figure 2 shows that only the public academic schools (PuAc-1 & PuAc-2) had students with a mathematics score of not passing (i.e., "1") from the previous year, which means that those students had repeated the ninth grade because they had failed. None of the students in PuAc-2 had a "5" as their previous year’s mathematics grade. It is also interesting that students in PuAc-2 showed a horizontal line of means across all the previous year’s grades. PrSc had a lower mean only for those students having a “2” as their previous year’s mathematics performance.

Figure 1. Boxplot display of algebra scores to school type.

Ayhan Kursat Erbas

29

Figure 2. Boxplot display of algebra scores by school within previous year math grades.

Discussion The major finding of this study was that a large portion of the variance in student algebra performance at the beginning of ninth grade could be explained, not by sophisticated time and resource consuming variables, but by easily measured variables. Three variables: school type, grade level, and previous year mathematics performance explained 54% of the variance in algebra performance. This result is consistent with previously reported research predicting algebra achievement using more sophisticated variables (see Flexer, 1984; Hanna, Bligh, Lenke & Orleans, 1969). It was also found that gender was not a good predictor of algebra performance. Although this result is different from what Flexer (1984) found, it is consistent with the findings of large scale international studies concerning the effect of gender on mathematics achievement.

Although females have been, historically, considered educationally disadvantaged, large scale studies reveal that the gender gap has been greatly reduced in formal education. According to Lubienski, 30

McGraw, and Strutchens (2004), similarities rather than differences exist between males and females in overall NAEP results. Even though it was statistically significant at the eighth and twelfth grade levels, achievement differences between males and females were less noticeable when compared with differences related to race/ethnicity and socio-economic status. On the international level, the International Association for the Evaluation of Educational Achievement (IEA) Third International Mathematics and Science Study (TIMSS) revealed that in most countries, girls and boys had approximately the same mean mathematics achievement at seventh and eighth grades (Beaton et. al., 1996). The differences in achievement that did exist in some countries tended to favor boys rather than girls; however, about three-fourths of the differences were not statistically significant, indicating that in most countries, gender differences in mathematics achievement are generally small or negligible in the middle years of schooling. Similarly, findings from TIMSS-Repeat revealed that the gender differences in mathematics achievement were negligible at the eighth grade level (Mullis, Martin, Gonzalez, & Chrostowski, Predicting Algebra Performance

2004). While girls significantly outperformed boys in a few countries, the opposite was observed in a few others, including the United States. In a more recent international study, Programme for International Student Assessment (PISA) revealed that gender differences favoring boys exist in mathematics achievement in some participating countries, but the differences tend to be small (OECD, 2004).

Even though the students in this study were at the same age levels, the effect of grade levels (i.e., “preparatory” and “9th grade”) were significant. This is probably because of the fact that students in 9th grade were introduced to algebra concepts (for example, functions) that differed from the ones they had encountered in the middle school algebra curriculum and the students had more experience in manipulating variables and literal expressions (in the traditional sense), whereas students in preparatory grade were usually given intensive instruction in a foreign language. However, I also observed that students in both grade levels did equally well in solving linear equations with a single unknown, which is required of all students in both seventh and eighth grade as a part of the curriculum.

Perhaps the most interesting result of this study was that school type was found to be the dominant predictor of students’ performance. Alone, it explained 33% of the variance in algebra performance. The order of students’ performance from highest to least was PrSc, VoTe, PuAc-1, and PuAc-2. This result raises important questions about the reasons for the variance or gaps in students’ performances in different school types. One simple explanation of the difference might be the higher quality of instruction, or education in general, in private schools. Because of the tuition and other private resources, these schools are able to hire experienced teachers and have greater flexibility and willingness (in both teacher and administration) to use alternative methods and materials in instruction. There is teacher and school accountability to the parents who pay for their children’s education, as the parents want to know their money is well spent. This is in contrast to the situation in public secondary schools, where the education is free, parent-school relations are weak, and accountability is virtually nonexistent.

Socio-economic levels of the students in the private and public schools are also noticeably different.

Parental education levels for the students in private schools tend to be higher, their attention to their children's education is more pronounced, and their methods for dealing with academic problems are

Ayhan Kursat Erbas

usually different, too. These factors tend to contribute to students’ motivation in a positive sense.

Both the private high school and the vocational high school have higher expectations and criteria for the students who apply to those schools after primary education, as is evident from the distribution of the students’ previous year mathematics score/grades. For example, approximately 80% of the students in PuAc-2 had previous year mathematics scores lower than “3”, which indicates that most of the students at that public school were average or below average. While this helps us to understand why school and previous year math performance explained 44% percent of the variation, such observations are perhaps indicative of the differences in the general characteristics of the students (for example, socio-economic status, mathematics achievement they carry from primary school to high school, etc.) we might find/observe in students of different types of schools. This larger framework of characteristics is perhaps the place to look for variables in the students’ algebra performances; and variables such as parental and teacher influence, teaching methods, and how students learn algebra in different school settings should be studied in detail. As is often the case, this study, which started with one question, has led to many others for future research.

REFERENCES Beaton, A. E., Martin, M. O., Mullis, I. V. S., Gonzales, E. J., Smith, T. A., & Kelly, D. L. (1996). Mathematics Achievement in the middle school years: IEA`s Third International Mathematics and Science Study. Chestnut Hill, MA: Center for the Study of Testing, Evaluation, and Educational Policy, Boston College.

Booth, L. R. (1984). Algebra: Children's strategies and errors.

Windsor, UK: NFER-Neslon.

Booth, L. R. (1988). Children's difficulties in beginning algebra. In A. F. Coxford (Ed.). The ideas of algebra, K-12 (1988 Yearbook of the National Council of Teachers of Mathematics) (pp. 20–32). Reston, VA: NCTM. Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297–334.

Education and Training Board. (1991). Ilkögretim (5 + 3 = 8) matematik dersi programi [Mathematics curriculum for primary education (5 + 3 = 8)]. Ankara, Turkey: The Turkish Ministry of National Education.

English, L. D., & Halford, G. S. (1995). Mathematics education: Models and processes. Mahwah, NJ: Lawrence Erlbaum.

Flexer, B. K. (1984). Predicting eighth-grade algebra achievement.

Journal for Research in Mathematics Education, 15, 352–360.

Fraenkel, J. R., & Wallen, E. N. (1996). How to design and evaluate research in education. San Francisco: McGraw-Hill.

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Green, S. B. (1991). How many subjects does it take to do a regression analysis? Multivariate Behavioral Research, 26, 499–510.

Hanna, G. S., Bligh, H. F., Lenke, J. M., & Orleans, J. B. (1969).

Predicting algebra achievement with an algebra prognosis test, Iqs, teacher predictions, and mathematics grades. Educational and Psychological Measurement, 29, 903–907.

Herscovics, N. (1989). Cognitive obstacles encountered in the learning of algebra. In S. Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of algebra (pp.

60–86). Hillsdale, NJ: Lawrence Erlbaum.

Howson, A. G. (1991). National curricula in mathematics. Leicester, UK: The Mathematical Association.

Huberty, C. J. (1994). A note on interpreting an R2 value. Journal of Educational and Behavioral Statistics, 19, 351–356.

Huberty, C. J., & Petoskey, M. D. (1999). Use of multiple correlation analysis and multiple regression analysis. Journal of Vocational Education Research, 24, 15–43.

Kieran, C. (1989). The early learning of algebra: A structural perspective. In S. Wagner and C. Kieran (Eds.), Research issues in the learning and teaching of algebra (pp. 60–86).

Hillsdale, NJ: Lawrence Erlbaum.

Kieran, C (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390–419). New York: Macmillan.

Küchemann, D. (1981). Algebra. In K. M. Hart (Ed.), Children's understanding of mathematics: 11–16 (pp. 102–119). London: John Murray.

Lubienski, S. T., McGraw, R., & Strutchens, M. (2004). NAEP findings regarding gender: Mathematics achievement, student affect, and learning practices. In P. Kloosterman and F. Lester (Eds.), Results and Interpretations of the 1990 through 2000 Mathematics Assessments of the National Assessment of Educational Progress (pp. 305–336). Reston, VA: NCTM. Mullis, I.V.S., Martin, M.O., Gonzalez, E.J., & Chrostowski, S.J. (2004). Findings from IEA’s Trends in International Mathematics and Science Study at the Fourth and Eighth Grades, Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College. Retrieved February 7, 2005, from http://timss.bc.edu/timss2003i/mathD.html National Council of Teachers of Mathematics. (1989). Principles and standards for school mathematics. Reston, VA: Author.

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

Organization for Economic Cooperation and Development (OECD). (2004). Learning for tomorrow’s world: First results from PISA 2003. Paris: OECD. Retrieved February 7, 2005, from http://www.pisa.oecd.org Pedhazur, E. J. (1997). Multiple regression in behavioral research.

Forte Worth, TX: Harcourt Brace.

Roth, P. L. (1994). Missing data: A conceptual review for applied psychologists. Personnel Psychology, 47, 537–560.

Whitman, B. S. (1976). Intuitive equation solving skills and the effects on them of formal techniques of equation solving (Doctoral dissertation, Florida State University, 1975).

Dissertation Abstracts International, 36, 518A. 1

. Brief information on the Turkish educational system is accessible in Appendix A. For a complete description of the statistical methods used, please see Appendix (C).

32

Appendix A: Turkish Educational System and School Types The Turkish educational system consists of pre-primary education (Pre-K), primary education (K1-8), secondary education (9-11) and higher education (universities and two year colleges).

For all levels of primary and secondary education, there are special education schools serving handicapped (exceptional) children. In Turkey, children ranging in age from 6 to 14 must attend primary school, which is free for all students.

Secondary education encompasses three years in the general high schools (public high schools, science schools, Anatolian high schools, etc.) or four years in the vocational and technical high schools (technical education schools, Anatolian technical education schools, religious education schools, commercial and tourism education schools). General high schools do not prepare students for a specific profession but rather prepare them for higher education. Vocational high schools train qualified individuals for various professions, and also prepare students for higher education. Anatolian high schools, private secondary schools, and some vocational and technical high schools require one year of English (German or French in a few cases) study before the students can continue their three year normal secondary schooling. Although mathematics courses are required in preparatory school, the weekly hours are fewer than in a regular ninth grade. In the sample of this study, there were students in preparatory class in the private high school and the vocational technical high school. However, the ninth grade students in the sample did not go through the preparatory school program.

At the end of eighth grade, on a single date towards summer, students take a non-compulsory centralized/standardized test called the High School Entrance Exam (or shortly LGS in Turkish). Usually the result of this test determines the types of schools students will attend in for their secondary education.

Science schools, Anatolian high schools, and most of the (Anatolian) technical education schools accept students with the prerequisite scores on the LGS. The acceptance policies of public high schools and private secondary schools vary. The higher the school’s perceived quality and reputation (and achievement), the more selective and difficult the acceptance process is. Public schools usually have no criteria for acceptance other than a primary school diploma. However, some public schools have a special division, Super High School, for students who are above average or high achievers. Students are accepted based on their LGS score and their primary school graduation GPA. Acceptance policies for the private schools vary, but a student’s primary school graduation GPA and LGS score, if it is available, again have deterministic roles. Furthermore, since private schools usually have expensive tuitions, socio-economic status is the single most important consideration for attending those schools.

There might be scholarships for a select few students.

The private school in the sample of this study was founded by the Foundation of National Education, which has several similar intuitions throughout the big cities of the country.

They accept students based on previous academic performance, primary school GPA, and ability to pay the tuition and other expenses. The two public high schools in the study were located in a socio-economically low to mid-level area of the district. They accept students with no particular criteria, except possibly a requirement that students live nearby. The vocational technical high school in the sample had a mixed group of students accepted based on their LGS score and then considered primary school GPA if they had space to accommodate more students.

Predicting Algebra Performance

Appendix B: Sample Items and information asked in the Algebra Test Information • Age • Gender • Date of birth • Previous year’s mathematics grade.

Items • Simplify 5(p + q) " 2(p + q) • When is the following inequality true? Always / Never / Sometimes ! o L+M+N=L+P+N • What is the area of this rectangle?

5 e

2

• •

Simplify

•

Simplify

• •

!

Factorize the following expression Solve for x: 3 = 5 !

• •

! expression (3x + 2) 2 so that there Rewrite/Expand the following are no parentheses. !

What is the value of a in the following equality? 69 " 96 = 37 !

•

If

"

• •

If If

0 "x " 2

!

•!

!

•

!

•

a +x b +x 1 2 + 3x x

2x + 1

(x " 1) 2 + 3(x " 1)

x

7"a

If

a 2

=

"b

a2 < a

3

=

"c 4

, list a, b, c from smaller to bigger.

! of x " y ? and 3 " y " 5 , what is the least value and a. b > b , would it be true that b < 0 ? Why?

# &2 %x " 5 ( ! = (x " 7) 2 , what is x? $ ' the!sum A = 2a + b + 3c , how

Appendix C: Statistical Information The school variable has four categories consisting of VoTe, PuAc1, PuAc-2, and PrSc, and scaled by creating three dummy variables. More precisely, PrSc, VoTe and Pac-1 were coded as 1 0 0 0, 0 1 0 0, and 0 0 1 0 consecutively in the data matrix. Here, it should be noted that the 0 0 0 row along those three columns indicates the coding for PuAc-2.

Analyses reported here had been done by using Summarize (descriptive), Regression (linear), Correlations, and Missing Value Analysis commands in SPSS 8.0 and by using the Proc Reg (best subset analysis) command in SAS 6.12.

Preliminary Data Analysis The data were first examined for missing values. The examination revealed that 17 subjects (about 8% of the sample) had one or two missing values. Those were determined by using the regression imputation technique (Roth, 1994). Before proceeding with a regression analysis, I searched for potential outliers using standardized residuals (Pedhazur, 1997, p. 44) and judged that nine students had Y scores and X-vector scores that were extensively distant from the Y-mean and X-vector of means. After those outliers were removed, the final data matrix was 208 × 5.

I accepted certain assumptions about the data matrix before the analysis. First, I assumed that any row vector of scores was independent of all other score vectors. Second, a normal probability plot was used to assess multivariate normality. For these data, the plot was virtually linear, indicating that the condition of normality was satisfied (Figures 3 and 4).

! !

In much A will increase if a is increased by 3, b is increased by 5, and c is increased by 8?

Rearrange the following expression for k so that you can find ! what k is: S = V " hk a

• •

•

What is the common solution for the following system of equation: 3b + 2a = 13 and 2b – 3a = 0. !

A ball bounces one-third of the height it is fallen. If it bounces 8 cm after it falls three times, what is the original height it falls the first time?

As shown in the following figure, the first pool in the fountain is filled by the water directly from the fountain and the rest two pools are filled by the water that drops from the one on the top after each is filled out. The volumes of the pools 1, 2, and 3 are v, 2v, and 3v respectively and the first pool fills out in 2 hours. If the fountain is left open for 10 hours, what percent of the third pools is filled with water? Why?

Figure 3. Histogram of standardized residuals for predictors of algebra performance

Ayhan Kursat Erbas

33

With the data assumptions met, I proceeded to determine the weights (b values) in the optimal linear composite of the six X variables (note that the school variable was converted into three dummy variables).

6

^ Y = b0 +

" b i Xi i =1

!

Additional analyses were carried out to determine if some analysis units (students) were having an extreme influence on the results.

Examining the Cook D index value (Pedhazur, 1997, p. 51), I concluded that no student had undue influence on the bias of the weights. Similarly, I concluded that no student score vector exhibited undue influence on the precision of weight estimates, since there were no extreme covariance ratio index values. On the other hand, examining DFBETA index values (Pedhazur, 1997, p.

52), I found student score vectors that unduly influenced the b weights (considered estimates of population counterparts).

Multiple Regression and Supplementary Analyses Figure 4. Normal cumulative probability plot of observed versus predicted standardized residuals for predictors of algebra performance.

The homogeneity of the variance in the criterion variable (i.e., Algebra Test) across the range of X-variable scores was assessed by examining the residual plot in Figure 5. Because of the elliptical shape of the plot, the homogeneity condition can be considered satisfied.

The regression equation for data was as follows: Y' = -30.823 -.735(Gender) +13.42(GradeLevel) + 4.401(MathPerformance) +21.267(PrSc) 14.514(VoTe) + 8.170(PuAc-1), with nonstandardized weights and, Y' = -.027(Gender) +.382 (GradeLevel) + .391(MathPerformance) + .577(PrSc) +.46(VoTe) + .289(PuAc-1), with standardized weights.

Note that to estimate scores for students in PuAc-2, one needs to enter 0 in place of PrSc, VoTe and PuAc-1 since it is coded as all zeros along three dummy variables.

To determine how well the resulting equation fit the data for the 208 students, the simple correlation between Y and Y' (i.e., R) was calculated. I found R2 â&#x2030;&#x2026; .55. For inferential purposes, one proceeds if this value is significantly larger than the value that would result by chance (Huberty, 1994). That is, is the R 2 value of .55 significantly greater than the chance value of p/(N-1) = 4/(2081) = .02? The statistical test yielded statistics for this analysis of F(8,201) = 24.08, p = .0001. Thus, I concluded that, overall, the predictions of Y using the four predictors were significantly better than chance.

To assess how well the derived equation would fit data based 2 , was calculated by on new samples, the adjusted R-squared, Radj the formula

R2

adj

=1 "

it has been found

N+p N"p

R2

adj

(1 " R 2 ) (see

= .52 .

Huberty & Petoskey, 1999) and

! size (i.e., how much better The effect

than !chance) question was addressed by considering the 2 " p /(N " 1) , which was approximately .49 for these difference Radj !

data. In general, it seems that the prediction rule is highly usable.

Figure 5. Residual plot (Y' vs. Y-Y') for predictors of algebra performance.

34

!

Predicting Algebra Performance

The Mathematics Educator 2005, Vol. 15, No. 1, 35–43

A Compendium of Views on the NCTM Standards Serkan Hekimoglu & Margaret Sloan Reactions to the publication of the National Council of Mathematics (NCTM) Standards, in both 1989 and 2000, revealed the extent to which differing sets of values and beliefs had divided the mathematical community.

This paper rejoins the debate surrounding the Standards, exploring some of the opposing points of view while offering some perspectives on the roots of the conflict, the current status of the debate, and some suggestions as to what might be done to foster a shared vision for the future of mathematics education. The authors illustrate how various beliefs about the key issues of today's mathematics education reform initiative might cloud the way the Standards are interpreted and implemented and argue that ongoing debate over the Standards should be considered a constructive means of exploring possible avenues for the reconciliation of conflicting ideas while providing safeguards against both an erosion of the reform effort and desultory implementation of the Standards.

The publication of the NCTM (1989) Standards touched raw nerves and created an extensive dialogue within and among many groups whose interpretations articulated multiple perspectives on the condition and purpose of mathematics education. More than simple analyses of the issues from opposing points of view, discussions about the Standards are often complex, with important educational, philosophical, social, and political dimensions. It is interesting to note that while some criticized the Standards for being too extreme (Cheney, 1997; Finn, 1993; Haimo, 1998; Wu, 1997), others argued the need for more radical changes (Apple, 1992). One of the aims of this paper is to examine objectively how different individuals and groups perceive the NCTM’s Standards (1989; 2000) and how these various perceptions might be consolidated into a cohesive force for change. The authors believe that what some may hear as a cacophony of voices can be experienced as a developing harmony if we are mindful of the contexts from which these various voices emerge and thereby Serkan Hekimoglu received his undergraduate and masters degree in Civil Engineering. He has taught a variety of engineering, math, stat, and physics courses. He is currently working toward his Ph.D. in Mathematics Education and Masters degree in Mathematics at The University of Georgia. He is interested in using technology in collegiate level teaching, applications of mathematics to real world problems, integrating physics and mathematics in a calculus class, and students' learning of calculus topics.

Margaret Sloan has a bachelor's degree in Accounting, MEd.and EdS. in Mathematics Education, and is currently working towasrd her PhD in Mathematics Education at the University of Georgia. Her research interests include the teaching and learning of mathematics at the secondary level, especially in rural communities.

Serkan Hekimoglu & Margaret Sloan

understand the evolutionary process behind the publication of the Standards.

A brief historical sketch Over the second half of the twentieth century, the nature of mathematics education in the U.S. changed dramatically as it passed through reincarnations based on New Math, Back to Basics, Problem Solving, and the NCTM Standards (NCTM, 1989; 2000). The driving forces and educational objectives of each initiative were different, and analyzing or interpreting these reform movements requires a consideration of the social and political climate in which each reform was initiated.

The first reform movement, New Math, was on the drawing board even before the launch of Sputnik in 1957, but the Soviet Union's scientific achievement in space created in America’s academic and political arenas a sense of immediacy about improving mathematics education in the United States.

Policymakers perceived a need for a new generation of highly qualified mathematicians, scientists, and engineers whose work would produce, among other things, a space program which would outpace that of the Russians. The New Math reform promised to decrease the gap between university mathematics and high school mathematics (Howson, Keitel, & Kilpatrick, 1981). A de facto shortcoming of the movement was the creation of a curriculum based solely on logical principles, dismissing the psychological dimensions of learning; but even more damaging was the lack of effective professional development for teachers who were to implement the program (Bass, 1994). Consequently, the movement’s ideals did not mesh well with classroom reality; and, 35

inevitably, there was a demand for changing the mathematics curriculum and instructional practices (Kilpatrick, 1992). The change came in the form of an almost complete reversal, a Back to Basics backlash.

In the 1970s, the Back to Basics movement was based on behavioral principles supported by the work of Thorndike and Skinner, and instructional strategies were focused on basic skills, emphasizing the mastery of computation (Howson, et al., 1981). Because teachers had been perceived as generally ill-equipped for the instructional demands of New Math, it was thought that well-designed instructional materials could overcome any shortcomings in teachers' content knowledge. What soon became evident in the Back to Basics movement was that there was no such a thing as a “teacher-proof” mathematics curriculum (Erlwanger, 1973), and the mathematical education community once again was faced with the challenge of developing a curriculum to bring effective mathematics instruction into the classroom.

In response to this need, NCTM initiated Problem Solving, an approach to mathematics instruction in which problem-solving techniques, within modified real-world contexts, would promote meaningful learning and teaching of mathematics (NCTM, 1980).

Although such problems had always been an integral part of the mathematics curriculum, this initiative considered the process of problem solving a vehicle for learning mathematics by encouraging students to develop logical reasoning skills and take responsibility for their learning (Stanic & Kilpatrick, 1989).

Throughout the 1980s, the importance of raising expectations for students, providing mathematics within a historical context, and demonstrating the usefulness of mathematical understanding became the focus of various research efforts. Publications such as A Nation at Risk (National Commission on Excellence in Education, 1983) and Educating Americans for the 21st Century (National Science Board Commission, 1983) documented the need for change and generated recommendations for mathematics education reform.

At the same time, mathematics education started to gain legitimacy with a proliferation of mathematics education research, produced by an ever-increasing number of scholars. With the emergence of constructivist theory as a dominant presence, there was a shift in mathematics education research from the investigation of teacher and student behaviors to an exploration of cognition and context (Cooney, 1994).

The NCTM Standards (1989) was born in this interesting arena, where researchers were beginning to identify vital components of learning and teaching 36

mathematics and in which remnants of New Math, Back to Basics, and Problem Solving were very much in evidence.

Policymakers also began to reaffirm the need for the educational sector to meet the needs of the economy (Glickman, 1998). Leaders of business and industry made their views known, especially as to the need for all students to be able to reason, design models, think creatively, and solve problems. There was a growing awareness that mathematics education was a part of the political structure and that mathematics educators could no longer be ignored during the formulation, debate, and implementation of policies and actions affecting mathematics instruction at all levels (Carl & Frye, 1991). With such empowerment, mathematics educators were motivated to better promote their discipline to both the public and policymakers (Crosswhite, 1990).

Concurrent with these developments, technological advances were being integrated into business and industry; and schools were expected to prepare students for the emerging information age. The utilization of computers was seen as a driving force for scientific and intellectual progress, with applications in most academic disciplines; and the new technologies contributed to the production of an expanding mass of mathematical knowledge (Bass, 2003; Hekimoglu, 2002). With this new capability came ontological and epistemological concerns and discussions about the process of mathematical proof (Bass, 2003; Ernest, 1998; Lakatos, 1998).

In the wake of the technological revolution, a quasi-empiricist and fallibilist philosophical stance based on the works of Gödel, Wittgenstein, and Lakatos emerged (Ernest, 1991). Mathematics should not be seen as a cornerstone of absolutism, especially in terms of its former identity as an objective and culture-free discipline. The existence of a multicultural society demanded awareness of differences in achievement in mathematics for African American, Hispanic, Native American, female, and low-income students (Moses, 1994; National Commission on Teaching and America’s Future, 1996; National Research Council, 1989). Many believed a redefinition was in order and that "equity in mathematics education" should be based upon enrichment, fairness, empowerment, and cultural diversity.

In the 1980s, NCTM tried to inform the general public about the perceived crisis in education and called for a significant departure from then-current practice in terms of content and pedagogy. The goal was to help the mathematics education community Views on the NCTM Standards

reach an accord with the needs of society and students, developments in mathematics and the application of mathematics, and the evolution of understanding about mathematics learning (NCTM, 1980; 1989). The 1989 Standards, designed to make mathematics accessible to all students, was startling to many mathematicians and mathematics educators, but it was primarily based on evidence uncovered by mathematics education researchers regarding what works, what does not work, and what could work more effectively and equitably (Hiebert, 1999; Research Advisory Committee, 1988).

What are the Standards? Who needs them?

Understanding and addressing concerns about the Standards requires a twofold approach: first, educators and mathematicians must agree that there is a need for having standards; and, second, they must have an awareness of, and an appreciation for, various perspectives on the issues addressed by the Standards.

Among the initial concerns was what could, or could not, be inferred from the name of the publication.

Generally, “standards” are used to establish measurable and ascertainable degrees of uniformity, accuracy, and excellence for products or services; and such “standards” are based on measurements, principles, and agreements which contain precise and static criteria, rules, and characteristics. Consequently, various objections were raised regarding the use of the term “standards” in the context of mathematics education (Les Steffe, in personal communication, 1012-2003) because in the context of the NCTM publication, the term is equivocal and overly broad.

Some questioned whether the NCTM Standards was merely a collection of slogans (Apple, 1992) or a welldefined statement about what mathematically literate students should know and be able to do: Was this document based on consistent philosophical and political stances about mathematics teaching and learning (Romberg, 1992; 1998)?

In the literature, individuals advocate using the Standards to reach an array of goals. Among those purposes: Elimination of vast differences in the quality of mathematics education (Delpit, 1995; Romberg, 1992); enhancing effectiveness of mathematics education by providing a clear focus for instruction, learning and assessment (Ravitch, 1995); demonstration of agreement and consensus (Labaree, 1984; O'Day & Smith, 1993); facilitation of the exchange of information (Noddings, 1992); establishment of a framework for school accountability (Apple, 1992; Labaree, 1984); confirmation of the legitimacy of mathematics education as a discipline by Serkan Hekimoglu & Margaret Sloan

steering the profession towards making investments in better prepared teachers (Darling-Hammond, 2003; Labaree, 1984); strengthening mathematics education by bridging academic levels, including higher education; guidance in the writing of mathematics curricula; communication to the public and policymakers about what students should know and be able to accomplish (Falk, 2000); and development of a national curriculum in response to international comparison studies of countries whose students have achieved excellence in mathematics.

Fully accommodating each of these various issues in a compendium of mathematics standards would be virtually impossible. Such a document would need to be pragmatic yet comprehensive, broad yet focused and effective, grandiloquent yet succinct. Furthermore, beyond the array of issues to be addressed, there lies an overwhelming assemblage of opinions on those issues.

Inevitably, the NCTM Standards (1989) was used for a variety of purposes, some of which were not in line with the developers' initial intentions of the publication being viewed as a resource guide (Romberg, 1992).

From the developers' points of view, the primary aim of the Standards was to describe a vision in which mathematics education would promote mathematical thinking by creating an awareness of the nature of mathematics, its role in contemporary society, its cultural heritage, and the importance of mathematics as an instrument and tool of learning (Romberg, 1992; 1998). NCTM made a credible and admirable attempt with the 1989 Standards, but as one might have expected, an intense political and philosophical debate, dubbed the math wars, began with the publication of NCTM (1989) Standards.

Critiques on NCTM Standards Appraisal of the NCTM Standards through bonafide and responsible critique depends on systemic considerations as well as specificity as to areas of interest, academic disciplines, and expectations. A comprehensive view of the NCTM Standards is needed to avoid focusing on any one of its components without regard to how that component fits into the overall scheme. It is not the authors’ intent to suggest that the viewpoints of mathematicians and mathematics educators are inherently dichotomous, or that mathematics educators at different levels have oppositional agendas, but it does seem that each group has a tendency to lose sight of common goals. It is the authors’ belief that understanding critiques of the Standards and using those critiques to initiate discourse about the issues may help build bridges 37

between the mathematics and mathematics education communities as well as define and address concerns within each group. These challenges can be successfully met if those involved in the debate will conform to a certain etiquette of criticism - articulating one’s position is not mutually exclusive to making an effort to understand, appreciate, and respect the merits of an opposing point of view.

Of course, many criticisms grew, not from misinformation or misinterpretation, but from differences in the opinions and beliefs of wellinformed mathematicians, mathematics educators, policymakers, parents, and others concerned about the future of mathematics education and how best to achieve an excellent educational environment. Some of these criticisms stemmed from interpretations about the purpose of decreased levels of attention or emphasis on particular skills in the curriculum without a clear understanding of the developers' initial message that the emphasis had to be shifted from students’ proficiency at learning skills to their understanding of the mathematics underlying those skills (Romberg, 1992, 1998). Some criticisms were spawned by the Standard's challenge to the tacit traditions of mathematics instruction at all levels.

In the NCTM (1989) Standards, one reader might see a focus on the process of mathematics learning, another reader observes the big ideas of teaching mathematics, and still another may describe or interpret the same items in terms of instructional strategies. Many individuals expressed negative judgments about the Standards and the direction in which mathematics education was headed (Ewing, 1996; Haimo, 1998). In particular, the NCTM Standards (1989), was severely criticized for its recommendations for a reduced emphasis on arithmetical computation and symbolic-manipulation skills; the need for teaching formal proofs (Cheney, 1997; Finn, 1993; Haimo, 1998; Roitman, 1998; Wu, 1997); the use of multiple assessment strategies; the integration of technology into mathematics instruction (Weiss, 1992); the de-emphasis of the abstract in favor of the concrete; and an emphasis on cooperative learning (Cheney, 1997; Haimo, 1998; Roitman, 1998; Wu, 1997).

The vision statement kindled a national discussion about the nature of the very heart of mathematics education, the need for defining literacy in mathematics. Defining mathematical literacy is not only dependent on one’s ontological beliefs about what mathematics is but also the epistemological beliefs regarding how it should be taught and the axiological 38

beliefs about where and how it is used. As defined in the NCTM Standards (1989), one who has become mathematically literate has become confident in one’s own ability to reason mathematically, has become a mathematical problem solver, and has learned to communicate mathematically. Following a decade of discussion and reflection, the 2000 NCTM Standards (2000) expands the definition of being mathematically literate to include having mathematical knowledge as a functional member of changing world: "Just as the level of mathematics needed for intelligent citizenship has increased dramatically, so too has the level of mathematical thinking and problem solving…." (p. 4).

What did the NCTM Standards (2000) say about skills and cooperative learning?

The NCTM Standards (1989) was criticized for advocating a reduction in the traditional emphasis on skills. The Standards message was that learning mathematics should not be limited to performing specific algebraic manipulation or basic arithmetic skills but should be expanded to include an in-depth understanding of concepts underlying these skills (NCTM, 1989; 2000). The debate on basic skills versus conceptual understanding goes back more than four decades (Brownell, 1956); and, although it might be reasonable to think that developing conceptual understanding might come at the expense of the development of basic mathematical skills (Roitman, 1998; Wu, 1997), the Standards aim was not to downplay the importance of basic skills. It was hoped that by providing students with an overall understanding of the role played by mathematics in their lives, students would be motivated to understand the mathematical concepts as well as master the skills.

In NCTM Standards (2000), importance of basic skills in the curriculum was underscored by statements such as the following: "Fluency with basic addition and subtraction number combinations is a goal for pre-K-2 years" (p. 84); and "when students leave grade 5, they should be able to . . . efficiently recall or derive the basic number combinations for each operation" (p.

149).

From the debate about basic skills vs. conceptual understanding, it became clear such terms as "basic skills" are not universally defined by any stretch of the imagination. Therefore, when interpreting the Standards as well as critiques of the Standards, a critical question arises: when people talk about basic skills, or any other issue, are they talking about the same thing? And why isn’t there more of a consensus about the meanings of these terms?

Views on the NCTM Standards

Prior to NCTM's Problem Solving initiative, basic skills were widely understood to be arithmetical computations and symbol manipulations. With advances in technology and access to global information and resources, “basic skills” have expanded to include data collection and analysis, measurement, and problem solving strategies; i.e., the use of logical reasoning and the application of basic algebraic and geometric concepts. The definition continues to be transformed, and there is little wonder that many criticisms have focused on the treatment of basic skills and their place in the curriculum.

Some might define working within a group as a basic skill. The Standards' support for the use of cooperative learning strategies was rooted in the developers' perception of classroom discourse as a requirement for the development of mathematical thinking and communication (NCTM, 1989; 2000).

The primary goals of cooperative learning include enabling students to take responsibility for their learning, to develop mathematical judgment, to lessen reliance on outside authority, and to promote mathematical communication. Implementation of cooperative learning strategies in the mathematics curriculum reflects the fact that many academic disciplines expect students to work cooperatively (Hekimoglu, 2003a) and believe the process of doing mathematics is an important form of social interaction (Dowling, 1998; Ernest, 1998). Some educators justify the use of cooperative learning strategies by reference to the traditional inquiry by future employers concerning a prospect’s ability to be a team player. The authors question, however, whether being a “team player” is in fact analogous to working effectively within a group and whether or not the overuse of cooperative learning strategies might inadvertently undermine the development of independent and individual learning styles. Participating in cooperative learning experiences might enable students to understand the mathematical aspects of society (Bishop, 1991), but the Standards did not suggest using it in every teaching period. Discussions about cooperative learning require the consideration of research findings that support why and when and how cooperative learning strategies should be implemented as well as pointing out possible pitfalls to avoid.

What did the NCTM Standards (2000) say about proof and technology?

Despite rumors to the contrary, the NCTM Standards (1989; 2000) never regarded mathematical proofs as outmoded or unimportant. Prior to Serkan Hekimoglu & Margaret Sloan

publication of NCTM Standards (1989), mathematical proof in the mathematics curriculum had become either nonexistent or had receded into meaningless ritual, perhaps as a result of the Back to Basics emphasis on skills. Students not only entered collegiate level mathematics courses without having an appreciation for the importance of mathematical proof but also without the skills and knowledge required for proof construction (Schoenfeld, 1987; Wu, 1994). One goal of the Standards was to provide an impetus for having students develop an understanding of informal proof through the use of heuristic arguments and explanations about their conjectures with the hope this process would lead students to the development and appreciation of formal mathematical proof (NCTM, 1989).

To accomplish this goal, the Standards advocated the use of technological tools as a method for demonstrating mathematical ideas and to help students generate hypotheses.

Contrary to common misinterpretations (Finn, 1993; Wu, 1997), the message was neither that the use of technological demonstrations should replace the need for proof nor that the construction of a mathematical proof should rule out the use of technology. The Standard's call for de-emphasizing formal mathematical proof was partly grounded in philosophical developments, based on the works of Gödel and Lakatos, questioning the historical gate-keeping role of mathematics, particularly the role of formal proof in the process of gaining acceptance by mathematicians (Ernest, 1991; Hersh, 1986). In response to the reactions and clarification about this issue stemming from the NCTM Standards (1989), the NCTM Standards (2000) clearly stated the need for doing mathematical proofs as a consistent part of students' mathematical experience: “By the end of secondary school, students should be able to understand and produce mathematical proofs" (p. 56).

What response does the mathematics community have to this recommendation? Mathematics education research has found that students do not share mathematicians' perceptions of doing proof as a backbone of doing mathematics and as way of doing mathematical research (Harel & Sowder, 1998; Selden & Selden, 2003). The necessity of proof for developing mathematical competency is not a universally accepted truth, and some math-based academic disciplines deemphasize mathematical proof (Hekimoglu, 2003a).

Additionally, studies have found that many pre-service teachers exhibit a skepticism about the relationship between proof and mathematical understanding (Hekimoglu, 2003b; Pandiscio, 2002). Furthermore, 39

where proof has been a traditional part of the curriculum, such as the use of two-column proofs presented in many high school geometry courses, there is an indication that the practice is not serving the educational purpose of teaching proofs (Herbst, 2002).

There seems to be a trend toward the use of informal and empirical arguments, as opposed to more formal mathematical proofs, in mathematical reasoning (Hekimoglu, 2003a; 2003b), indicating that formal proof remains of questionable value outside the mathematics research community. The inclusion of instruction about formal mathematical proof is often advocated as a necessity for helping students develop mathematical maturity; but exactly how doing mathematical proofs contributes to mathematical understanding is a relatively unexplored area. For many, successfully learning the process of formal proof construction requires an arsenal of previously developed logic skills rather than having one's reasoning skills developed by the process of learning to construct proofs. In the next revision of the Standards, it seems there is a need for clarification about the following critical questions: what is a proof, in what way is the process of constructing mathematical proofs an efficient way to teach logical reasoning, and how can mathematical proof be presented so as to engender a spirit of mathematical curiosity.

Another criticism of the NCTM Standards (1989) was centered on the didactic nature of technology integration. The Standards did not suggest that technology would be a magic bullet, correcting all the ills of the past or serving as a replacement for skill development (Romberg, 1992). The Standards position was based on the premise that technology was essential in the teaching and learning of mathematics and that technology influenced how and what mathematics might be taught as a result of the widespread impact of technology on society (Hansen, 1984; Kaput, 1992). In regard to the 2000 Standards, seeing technology simply as a tool for instruction would not only reflect a limited view of technology but would also undermine the power of technology available to both the public and mathematics education community. Furthermore, the commitment to the integration of technology into the curriculum at all levels was supported by mathematics education research findings suggesting technological tools were being used for a variety of different purposes ranging from computational assistance to intelligent tutorials to a medium for exploration and discovery.

It seems something of a mystery that many mathematicians object to the use of technology in 40

mathematics instruction when technology is commonly used to conduct research and communicate findings.

An inability to understand how technological tools can be used to solve mathematics problems often produces a dissatisfaction with students' mathematical achievement and their ability to use technological tools to solve problems in other disciplines (Hekimoglu, 2003a). With so many different messages being expressed by professors in method courses and mathematics classes about the value, or lack thereof, and the use, or misuse, of technology in mathematics education, pre-service teachers are being asked to negotiate a quagmire as they develop opinions and beliefs about teaching with technology (Leatham, 2002). Clearly, the use of technology in the mathematics curriculum is a complex issue involving the interplay of multiple factors, and clarification of these issues requires careful examination of research findings that shed light on the key components of successful technology integration.

And now what?

Perhaps the Standards greatest contribution to the mathematical community has been its role as a stimulus for more than twenty years of nonstop debate about these issues. During that time, virtually every aspect of mathematics education has been examined and discussed and debated in various arenas even beyond the confines of academia. Journalists and policymakers have raised a number of questions about mathematics, mathematics education, and the relationship between them. Mathematics educators are learning to communicate their position more clearly, especially on controversial issues such as basic skills, technology, and cooperative learning.

Philosophical and epistemological concerns about mathematics and mathematics education need to be publicly acknowledged. Constructivist pedagogy, for example, has often been misunderstood, perceived as a patchwork philosophical stance (Howe, 1998; Klein, 1997; Cheney, 1997). Mathematicians need to find ways to reconcile the inherent conflict between the formalist or Platonist view of the nature of mathematics (Ernest, 2000; Santucci, 2003; Schechter, 1998) and the constructivist pedagogy reflected in much of the Standards. Furthermore, beyond personal preferences and beliefs, society demands that why, how, and what mathematics should be taught depend on "simultaneous objective relevance and subjective irrelevance of school mathematics to one's set of values" (Ernest, 2000, p. 3).

Views on the NCTM Standards

Mathematicians and mathematics educators need to communicate with the general public, as well as with each other, about the primary goals of teaching mathematics and why these goals cannot be centered around the "high call" of promoting mathematical growth and/or creating mathematicians, although those are certainly worthy goals (Noddings, 1993; Woodrow, 1997). While mathematicians might express their desire that "students in schools should be taught abstract mathematical procedures through repeated practice of the procedures, in order that they reach the university conversant in the range of methods that they will need to use and apply there" (Boaler & Greeno, 2000, p.188), mathematics educators are keenly aware that many students will not be doing college level mathematics and that the diverse historical, social, philosophical, and psychological dimensions of mathematics learning and teaching cannot be ignored.

In regard to this issue, educators need to articulate why secondary mathematics education cannot be based solely on preparing students for collegiate level when there is a perceived need to produce a critical future supply of highly qualified mathematicians (Lutzer & Maxwell, 2003). Still, it is important to help collegebound students make a smooth transition from high school to college, and more dialogue between mathematicians and mathematics educators is needed to connect high school and college mathematics curricula and instructional strategies (Adelman, 1999).

The debates around the NCTM Standards might be beneficial for resolving conflicts about what students need to know to be prepared for college as well as what they need to know to become productive members of society.

Even though mathematics education research at the collegiate level is still in an embryonic stage, it has generally supported the Standard's stance on such issues as making connections between classroom mathematics and physical world applications to promote the development of conceptual understanding (Arney & Small, 2002; Karian, 1992; Solow, 1994).

Moreover, the Standard's vision of technology has been re-enforced in the collegiate mathematics community by The Mathematical Association of America (MAA) with its presentations of research findings in documents such as Computers and Mathematics (Smith, Porter, Leinbach, & Wenger, 1988) and The Laboratory Approach to Teaching Calculus (Leinbach, Hundhausen, Ostebee, Senechal, & Small, 1991). The importance of implementing cooperative learning strategies in undergraduate mathematics classes has also been echoed in the Serkan Hekimoglu & Margaret Sloan

Cooperative Learning in Undergraduate Mathematics (Rogers, Reynolds, Davidson, & Thomas, 2001) and A Practical Guide to Cooperative Learning in Collegiate Mathematics (Hagelgans et al., 1995).

The development of the next revision of the Standards demands that mathematics educators consider how the current provisions have actually been implemented (Roitman, 1998) and what overall systemic changes could be made to promote the success of the reform movement. Such a goal requires a collaboration between mathematicians and mathematics educators through discussion and delineation of the issues surrounding systemic changes in secondary education. Careful consideration of existent research findings on teaching and learning mathematics as well as the further development of specific research methods and theoretical frameworks, drawn from fields of study such as anthropology, sociology, psychology, linguistics, and philosophy may enable educators to question and explore the nature of mathematics, mathematics teaching and learning, and school structures.

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Serkan Hekimoglu & Margaret Sloan

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CONFERENCES 2005…

AMESA Eleventh Annual National Congress

Kimberley,South Africa

June 27–30

Melbourne, Australia

July 10–15

ICTMT7 7th International Conference on Technology in Mathematics Teaching

Bristol, UK

July 26–29

JSM of the ASA Joint Statistical Meetings of the American Statistical Association

Minneapolis, MN

August 7–11

Shanghai, Nanjing, and Hangzhou, China

August 7–12

Prague, Czech Republic

August 21–26

Rock Eagle, GA

October 20–22

Roanoke, VA

October 20–23

Fort Worth, TX

November 10– 12

Johor Bharu, Malaysia

November 25– December 1

Taichung, Taiwan

December 14– 18

Tampa, FL

January 26–28, 2006

San Francisco, CA

April 8–12, 2006

St. Louis, MO

April 26–29, 2006

http://academic.sun.ac.za/mathed/AMESA/Index.htm PME-29 International Group for the Psychology of Mathematics Education http://staff.edfac.unimelb.edu.au/~chick/PME29/

http://www.amstat.org/meetings/jsm/2005/ ICMI – EARCOME3 The Third East Asia Regional Conference on Mathematics Education http://euler.math.ecnu.edu.cn/earcome3/Announcements.htm SEMT International Symposium Elementary Mathematics Teaching http://www.pedf.cuni.cz/kmdm/aktivity/semt.htm GCTM Georgia Council of Teachers of Mathematics Annual Conference http://www.gctm.org/georgia_mathematics_conference.htm PME-NA North American chapter International Group for the Psychology of Mathematics Education http://pmena.org SSMA School Science and Mathematics Association http://www.ssma.org 8th Internal Conference of The Mathematics Education into the 21st Century Reform, Revolution and Paradigm Shifts in Mathematics Education http://www.sigrme.org/announce/MalaysiaFA.htm AMS First Joint International Meeting with the Taiwanese Mathematical Society http://www.ams.org AMTE Association of Mathematics Teachers http://amte.net AERA American Education Research Association http://www.aera.net NCTM National Council of Teachers of Mathematics http://www.nctm.org

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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 tme@coe.uga.edu. 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 tme@uga.edu 105 Aderhold Hall The University of Georgia Athens, GA 30602-712

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About the cover David Dorcas, who is currently taking Euclidean geometry, is a sophomore at North Oconee High School in Georgia.

After watching the movie Pi*, he became intrigued with the golden spiral and did some research about it on his own.

Using the tools of the computer program GeoSketchpadTM, David was able to create the golden spiral, along with other mathematically based drawings. The piece featured on the front of this issue is one of Davidâ&#x20AC;&#x2122;s sketches in which the ratio of consecutive segments is the golden mean. Below is his explanation for creating the golden rectangle, which he used to create his drawings.

B First create a square.

Then find the midpoint (M) of the base.

Next create a circle with M as your center and A as your outer edge.

Extend line CD until it interseccts with the circle (point E).

Create a perpendicular line at point E. Extend line BA until it intersects that perpendicular line (point F).

Rectangle BFEC is the golden rectangle.

C

A

M

D

F

E

*Watson, E. (Producer), & Aronofsky, D. (Director). (1998). Pi. [Motion picture]. United States: Artisan Entertainment.

More mathematical artwork by David Dorcas:

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Artwork by David Dorcas continued:

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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 Georgiaâ&#x20AC;&#x2122;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.coe.uga.edu/tme

TME Subscriptions TME is published both online and in print form. The current issue as well as back issues are available online at http://www.ugamesa.org, then click TME. A paid subscription is required to receive the printed version of The Mathematics Educator. Subscribe now for Volume 16 Issues 1 & 2, to be published in the spring and fall of 2005.

If you would like to be notified by email when a new issue is available online, please send a request to tme@uga.edu To subscribe, send a copy of this form, along with the requested information and the subscription fee to The Mathematics Educator 105 Aderhold Hall The University of Georgia Athens, GA 30602-7124

___ I wish to subscribe to The Mathematics Educator for Volume 16 (Numbers 1 & 2). ___ I would like a previous issue of TME sent. Please indicate Volume and issue number(s): ___________________

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Amount Enclosed ________________ subscription: $6/individual; $10/institutional each back issue: $3/individual; $5/institutional

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In this Issue, Guest Editorial… The Role of Mathematics Instruction in Building a Socially Just and

Diverse Democracy DEBORAH LOEWENBERG BALL, IMANI MASTERS GOFFNEY, & HYMAN BASS The Paradigm Shift in Mathematics Education: Explanations and Implications of Reforming Conceptions of Teaching and Learning MARK W. ELLIS & ROBERT Q. BERRY III Integrating Technology Into the Mathematics Classroom: The Role of Teacher Preparation Programs REGINA M. MISTRETTA Predicting Turkish Ninth Grade Students’ Algebra Performance AYHAN KURSAT ERBAS A Compendium of Views on the NCTM Standards SERKAN HEKIMOGLU & MARGARET SLOAN

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