____ THE _____ MATHEMATICS ___ ________ EDUCATOR _____ Volume 13 Number 2
MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA
A Note from the Editor
Editor Brian R. Lawler
Along with a team of editors, reviewers, and other colleagues that help me to think, I wish to present to you—the ever-present reader in our work—the second and final issue to be produced during my brief tenure as editor of Volume 13 of The Mathematics Educator. At this moment in mathematics education, while reform remains a normalizing discourse and accountability snarls as a threatening tyrant, the work of teachers clearly remains central in our collective efforts to grow the field. In these welcoming words, I hope to interest you in the papers assembled here, while provoking you to read alongside ideas and theory that may not usually be with you. In this issue’s final essay, Adelyn Steele, a Kansas state finalist for the Presidential Award of Excellence in Mathematics and Science Teaching, reflects on the work of the teacher. She makes evident that this work, in which she suggests that we “just get out of the way,” is an immeasurably artistic maneuvering amid an intent to provoke both thoughtfulness and autonomy in the learner. Several of the papers herein review and build theory that can inform and impact our collective efforts toward such goals in teacher education. Boris Handal, Drew Ishii, and Norene Lowery review and build ground-level theory to help us think and act when working alongside evolving teachers. These researchers’ work for reform in mathematics education concentrates on pressing together the beliefs and actions of teachers, then developing reflective practitioners to make sense of what they do and are trying to do. Melissa DeHaven and Lynda Wiest help us consider reform by documenting effects of a particular design for a girls mathematics and technology program. Continuing to move in reverse order through the journal, Danny Martin opens up and troubles calls for equity in the reform discourse. Refusing the co-optation of equity work into the always already unjust institution of public education, he presents a view upward and into the inequitable structures of schooling to make possible the chance to think differently about our work for reform in mathematics education. In considering the body of research and theory within this issue of TME, it is evident the work of mathematics educators is intra-human relations and activity—and thus by it’s nature, political. Invigorated to know our work is not value-free, Brian Greer and Swapna Mukhopadhyay challenge us to think deeply about what is mathematics education for? What may emerge in our field if we reject Chomsky’s identified goal of schooling “to keep people from asking questions”? I hope this issue of TME is both insightful and stimulating. I hope the research and reference materials provide room for you, the reader, to think. I hope you are reminded to not stop wondering along with me, “What kind of politics am I doing in my classroom?”
Associate Editors Holly Garrett Anthony Dennis Hembree Zelha Tunç-Pekkan Publication Laurel Bleich Advisors Denise S. Mewborn Nicholas Oppong James W. Wilson
MESA Officers 2003-2004 President Dennis Hembree Vice-President Erik Tillema Secretary R. Judith Reed Treasurer Angel Abney NCTM Representative Holly Garrett Anthony Undergraduate Representative Tiffany Goodwin
Brian R. Lawler 105 Aderhold Hall The University of Georgia Athens, GA 30602-7124
About the cover Excerpt from a speech given at Western Michigan University, December 18, 1963: Some time ago, it was our good fortune to journey to that great country known as India. I never will forget the experience. I never will forget the marvelous experiences that came to Mrs. King and I as we met and talked with the great leaders of India, met and talked with hundreds and thousands of people all over the cities and villages of that vast country. These experiences will remain dear to me as long as the chords of memories shall linger. But I must also say that there were those depressing moments, for how can one avoid being depressed when he sees with his own eyes millions of people going to bed hungry at night? How can one avoid being depressed when he sees with his own eyes millions of people sleeping on the sidewalks at night, no beds to sleep in, no houses to go in. How can one avoid being depressed when he discovers that out of India’s population, more than 400,000,000 people, some 380,000,000 earn less than ninety dollars a year. Most of these people have never seen a doctor or dentist. As I notice these conditions, something within me cried out, “Can we in America stand idly by and not be concerned?” Then an answer came, “Oh, no, because the destiny of the United States is tied up with the destiny of India and every other nation.” I started thinking about the fact that we spend millions of dollars a day to store surplus food. I said to myself, I know where we can store that food free of charge, the wrinkled stomachs of the millions of God’s children that go to bed hungry at night. All I’m saying is simply this, that all life is interrelated, that somehow we’re caught in an inescapable network of mutuality tied in a single garment of destiny. Whatever affects one directly affects all indirectly. For some strange reason, I can never be what I ought to be until you are what you ought to be. You can never be what you ought to be until I am what I ought to be. This is the interrelated structure of reality. John Donne caught it years ago and placed it in graphic terms. “No man is an Island, entire of itself; every man is a piece of a Continent, a part of the main.” He goes on toward the end to say “Any man’s death diminishes me because I am involved in mankind; and therefore never send to know for whom the bell tolls; It tolls for thee.” It seems to me that this is the first challenge. This emerging new age. Dr. Martin Luther King, Jr.
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
Volume 13 Number 2
Table of Contents 2
Guest Editorial… What is Mathematics Education For? BRIAN GREER & SWAPNA MUKHOPADHYAY
Hidden Assumptions and Unaddressed Questions in Mathematics for All Rhetoric DANNY BERNARD MARTIN
23 The Fourth “R”: Reflection NORENE VAIL LOWERY 32 Impact of a Girls Mathematics and Technology Program on Middle School Girls’ Attitudes Toward Mathematics MELISSA A. DEHAVEN & LYNDA R. WIEST 38 First-Time Teacher-Researchers Use Writing in Middle School Mathematics Instruction DREW K. ISHII 47 Teachers’ Mathematical Beliefs: A Review BORIS HANDAL 58 In Focus… Just Get Out of the Way ADELYN STEELE 22 Upcoming conferences 60 Subscription form 61 Submissions information
© 2003 Mathematics Education Student Association All Rights Reserved
The Mathematics Educator 2003, Vol. 13, No. 2, 2–6
Guest Editorial… What is Mathematics Education For? Brian Greer Swapna Mukhopadhyay This is a great discovery, education is politics! After that, when a teacher discovers that he or she is a politician, too, the teacher has to ask, What kind of politics am I doing in the classroom? -Paulo Freire We are not experts in social and political theory, but rather educators struggling to understand the implications and manifestations of Paulo Freire’s (Freire & Shor, 1987, p. 46) statement, with particular reference to mathematics education. The views expressed here are personal and emergent, and intended to be provocative. According to Apple (2000): “It is unfortunate but true that there is not a long tradition within the mainstream of mathematics education of both critically and rigorously examining the connections between mathematics as an area of study and the larger relations of unequal economic, political, and cultural power” (p. 243). However, there are signs of change, building on a major shift within the discipline of mathematics education from a mainly cognitive and pedagogical perspective towards recognition of the historical, cultural, and social contexts of both mathematics and mathematics education (e.g., various chapters in Boaler, 2000). This shift is encapsulated in the phrase “mathematics as a human activity” whence the acknowledgment of the political situatedness of mathematics education is a natural outgrowth (Mukhopadhyay & Greer, 2001). Brian Greer is in the Department of Mathematics and Statistics and the Center for Research on Mathematics and Science Education at San Diego State University. His PhD was in the School of Psychology, Queen’s University, Belfast, where he worked until 2000. His interests have unfolded from a narrow focus on cognitive aspects to recognition of the cultural embedding and political ramifications of mathematics education. Swapna Mukhopadhyay is in the School of Education at Portland State University. For her PhD in Education at Syracuse University she studied the ethnomathematics of potters and weavers in a village in her native India. She characterizes mathematics as a cultural construction and is committed to the view that mathematics education should promote social justice. She serves on the committee of the Portland branch of Rethinking Schools.
Mathematics, Mathematics Education, And Mathematics In People’s Lives The answer to the question “What is mathematics?” is generally considered to be relatively unproblematic, although continuing to evolve as a result of internal developments and external factors such as accessibility to the power of computers as processors of symbols and images. However, having accepted that modern mathematics is a worldwide, unified discipline, close-knit through global communication and networks of scholars and institutions, there remain the questions of the relationship between that body of knowledge and what is taught in schools, how, and why. Given the pace at which mathematics has been, and is being, developed, the gap is increasing between the body of knowledge and what can reasonably be included in school education. At the same time, there is more and more concern about the gap between school mathematics and the lived experience of students and the adults that they become. Davis and Hersh (1981, pp. 39) composed an imaginary dialogue between “the ideal mathematician” and the public information officer of the University, part of which goes like this: P.I.O.: Do you see any way that the work in your area could lead to anything that would be understandable to the ordinary citizen of this country? I.M.: No. P.I.O.: How about engineers or scientists? I.M.: I doubt it very much. P.I.O.: Among pure mathematicians, would the majority be interested in or acquainted with your work? I.M.: No, it would be a small minority.
What is Mathematics Education For?
It is, then, hardly controversial to assert that mathematics is now too big to allow school students to be exposed to more than a fraction. So, on what basis, and by whom, are selections made? Both reflecting and reinforcing the highly organized nature of mathematics as a discipline, there is a very striking uniformity of school mathematics curricula across the world. Usiskin (1999, p. 224) observed students in Shanghai solving Euclidean geometry problems exactly like those in Japanese and American texts, even to the point of noticing the abbreviation “SAS” (for the Side-Angle-Side congruence condition) among the Chinese characters. As Usiskin also pointed out, the existence of international exercises such as the Third International Mathematics and Science Survey (TIMSS) assumes enough curricular commonality to make such comparisons meaningful (e.g., p. 213). As another example of how mathematics education shows remarkable uniformity over time and cultures, Fasheh (1997) describes living through four educational systems in Palestine—British, Jordanian, Jordanian with Israeli modification, “Palestinian”—and then comments: What is startling about the math curriculum is—with the exception of some changes at the technical level—how stubborn and unchanging it has remained under the four completely different realities in which I have lived, studied, and taught; how insensitive and unresponsive it has been to the drastic changes that were taking place in the immediate environment! When something like this is noticed, it is only natural to ask whether this is due to the fact that math is neutral or that it is actually dead! (p. 24)
However, in various parts of the world, attempts have been made to combat global homogenization of mathematics education combined with the predominant mode of teaching that dissociates mathematics from people’s lived experience. Describing “People’s Mathematics” (Julie, 1993), Volmink (1999) commented that it “developed independently and indigenously rather than an attempt to embrace the loudest fad from the West” (p. 94) and listed as distinguishing features: • an ability to reveal how school mathematics can be used to reproduce social inequalities • a rejection of absolutism in school mathematics and its contribution towards seeing mathematics as a human activity and therefore necessarily fallibilist • an incorporation of the social history of mathematics into mathematics curricula Brian Greer & Swapna Mukhopadhyay
• a belief in the primacy of applications of mathematics The last point above exemplifies one trend that we consider potentially positive. Insofar as there is greater emphasis in curricula on applications of mathematics and increasing incorporation of data handling into the curriculum, these changes open up possibilities for diversification through using mathematics to analyse socially and culturally relevant problems. For example, Gutstein (2003) writes about teaching in a low-income Mexican immigrant community in Chicago: I use ideas of social justice along with helping students develop mathematical power (being able to reason and communicate mathematically, develop their own mathematical thinking, and solve real-world problems in multiple and novel ways)—and pass the “gatekeeping” standardized tests. (p. 35)
Another ways of making connections between mathematics education and the lives of students is to break down the barriers between schools and communities (e.g., Abreu, 2002; Civil, 2002; Moll & Greenberg, 1992). Fasheh (2000) declared “I cannot subscribe to a system that ignores the lives and ways of living of the social majorities in the world; a system that ignores their ways of living, knowing and making sense of the world” (p. 5). By an extension of these principles, those who research mathematics education are separated only artificially from the social and political realities within which they work (Vithal & Valero, 2001). To summarize, we are suggesting that the relationships between these three aspects—mathematics as a discipline, mathematics as a school subject, and mathematics as a part of people’s lives—need serious analysis. I see mathematics playing an important role in achieving the high humanitarian ideals of a new civilization with equity, justice, and dignity for the entire human species without distinction of race, gender, beliefs and creeds, nationalities, and cultures, but achieving these goals depends on our understanding of the relation between mathematics and human behavior. Consideration of this relation is normally untouched by mathematicians, historians of mathematics, and mathematics educators. (D’Ambrosio, 1999, p. 143)
What Is Mathematics Education For? We list, and make brief comments on, a number of answers. All have validity, so how they are evaluated is a matter of balance and priorities, which vary with 3
experience, intellectual history, beliefs, values, and ideologies. 1. For some mathematicians, the obvious purpose of mathematics education is to produce more mathematicians (and also scientists, engineers, and others who will use substantial technical mathematics in their work). At the extreme, this supports a conception of mathematics education as a pyramid, with curriculum planned primarily for the few at the peak, and the majority left to struggle up as far as they can manage. Some of the calls for “mathematics for all” amount to just trying harder to push more people further up the pyramid. There is a hint of that attitude in the following statement by the National Council of Teachers of Mathematics (NCTM): NCTM challenges the assumption that mathematics is only for the select few. On the contrary, everyone needs to understand mathematics. All students should have the opportunity and the support necessary to learn significant mathematics with depth and understanding. There is no conflict between equity and excellence. (2000, p. 5)
2. However, recently it has become extremely common to portray the main reason for mathematics education as the training of a workforce able to compete successfully in the global economy of the information age. The following statement comes from a spokesperson of the People’s Republic of China, but could have come from almost anywhere in the world: As the economy adapts to information-age needs, workers in every sector must learn to interpret computer-controlled processes. Most jobs now require analytical rather than merely mechanical skills. So most students need more mathematical ability in school as preparation for their future jobs. ... [P]eople must deal daily with profit, stock, market forecast, risk evaluation etc. Therefore, mathematics relevant to these economic activities, such as ratio and proportion, operational research and optimization, systematic analysis and decision theory, etc., should be a part of school mathematics education. (Er-sheng, 1999, p. 58)
Gatto (2003) presents an argument that public schooling in the United States was shaped by industrialists (notably Carnegie, Morgan, Rockefeller, and Ford) in order to produce a docile and efficient workforce. 3. Briefly and uncontroversially, mathematics—as much as literature or music—is part of the cultural heritage that can make people intellectually well rounded and creative solvers of intellectual problems. We assert, without argument or evidence, that 4
mathematics education has mostly failed disastrously in these respects. 4. Mathematics is also characterized as the purest form of reasoning, embodying the highest standards of proof; and as a training in dispassionate, objective, rational thinking. We do not attempt here to analyse the various critiques of this position. 5. It is often stated that mathematics is needed as preparation for the practicalities of everyday life. Does this statement bear scrutiny? Is it not the case that most people handle the practicalities of daily life effectively without benefit of school mathematics beyond simple arithmetic and that the knowledge and skills that are essential are acquired through learning within practices situated outside of school? On the other hand, we argue below that there are other aspects of people’s lives that could and should be radically improved through access to mathematical tools for critical analysis. 6. From the perspective of a different value system, the most important reason for mathematics education is to make accessible to many people powerful mathematical ideas as conceptual and critical tools to analyse issues relevant to their lives (e.g., Skovsmose & Valero, 2002). For example, the application of mathematics as a critical tool for the analysis of American society is illustrated by an exercise beginning with the question “If Barbie was as tall as one of us, what would she look like?” (Mukhopadhyay, 1998). 7. According to Davis and Hersh (1986): The social and physical worlds are being mathematized at an increasing rate. The moral is: We’d better watch it, because too much of it may not be good for us. (p. xv)
Mathematics not only reflects our view of the world, but also helps to shape it, so that “when part of reality becomes modeled and remodeled, then this process also influences reality itself” (Skovsmose, 2000, p. 5). What Skovsmose terms “the formatting power of mathematics” is by no means a new development, but it is amplified by technological developments. It seems clear that the ratio: accessible information –––––––––––––––––––––––––––––––––– conceptual means for making sense of it is accelerating, with unforeseeable consequences. Looking Around In sketchy and illustrative form, some prominent features of the contemporary politico-educational scene in the USA are the following: What is Mathematics Education For?
Underfunded mandates: A recent entry in Harper’s Index (Sept. 2003) reads: Change since last year in federal spending to implement the No Child Left Behind Act: $1,200,000,000. (p. 13)
In case you are wondering, the change was downwards. A “black box” model for control of schools: Within the black box are the teachers and the students and the human interactions that constitute teaching/learning. What is inside the box can be ignored as control is exerted through the manipulation of external levers—money, testing, and punishment being the main ones. Corruption: Robert Kimball, an assistant principal in Houston, was surprised that in his high school with a freshman class of 1,000 that was reduced to fewer than 300 by senior year, the number of dropouts reported was zero (Winerip, 2003). When he blew the whistle, Robert Kimball was isolated and expects to be fired in January—you might like to track his story. Horn and Kincheloe (2001) compiled a generally skeptical analysis of the “Texas miracle”. Fantasy: The first President Bush set a goal for the USA to be number one in math and science education by the year 2000. Now it is mandated that every child in the USA will pass reading and math proficiency tests by 2014. There is only one way in which this could happen, namely by disappearing those who don’t make it, like the dropouts of Houston. Inequity: All of the above are contributory factors to the failure to diminish the “performance gap” between white students and minority groups, in particular African-Americans and Latinos. We attended a meeting recently where a public school teacher spoke of a report (in English) being sent to parents who do not speak English telling them that their child, who also does not speak English, had scored zero in a test written in English. There is currently a class action suit, Williams vs. the State of California, arguing that California provides a fundamentally inequitable education to students based on wealth, and based on language status. As background for this case, Gándara, Rumberger, Maxwell-Jolly, and Callahan (2003) have documented seven aspects of this inequity. Naïve expectations about the power of research: Slavin (2002) wrote as follows: At the dawn of the 21st century, educational research is finally entering the 20th century. The use of randomized experiments that transformed medicine, agriculture, and technology in the 20th Brian Greer & Swapna Mukhopadhyay
century is now beginning to affect educational policy… [A] focus on rigorous experiments evaluating replicable programs and practices is essential to build confidence in educational research among policymakers and educators. However, … there is still a need for correlational, descriptive, and other disciplined inquiry in education. (p. 15)
The best cautionary rejoinders that we know of to the expectations that mathematics education can be automatically improved through evidence-based policies generated by rigorous research (again, the image of a black box comes to mind) are Freudenthal’s (1978) book Weeding and sowing and Kilpatrick’s (1981) paper The reasonable ineffectiveness of research in mathematics education. The latter, in particular, points out that the improvement of mathematics education is hard because it is not an engineering problem, but a human problem. The endeavor rests in fundamental ways on questions that lie beyond the powers of research to generate definitive answers, but rather related to beliefs, values, and the aims of education. Intellectual child abuse: Without singling out any example (you might like to select your own), we assert that the most salient features of most documents that lay out a K-12 program for mathematics education is that they make an intellectually exciting subject boring. Emotional child abuse: One of the really big questions in mathematics education is: “Why do so many people fear and dislike mathematics?” Here is one answer, from a Bronx school (Wilgoren, 2001): It is a morning ritual… [The teacher] stalks across his classroom, scowls at his sixth-grade students and barks the same simple question: “What is this?” “This is math,” they respond. “I don’t have to like it to pass it. I don’t have to enjoy it to learn it. I don’t have to love it to understand it. But I must, and I will, master it”. (p. A1)
Final Comments Freire used the term “conscientization” to refer to a process of critical self-consciousness. As stated in the opening quotation, this implies reflection on the political nature of what we are doing as teachers or others engaged in education. During a recent meeting with students at Portland State University, Donaldo Macedo commented on the virtual absence from university education courses of classes on topics such as “ethics” or “ideology”. This comment recalls the statement of Chomsky (2000) that “the goal [of schools] is to keep people from asking questions that matter about important issues that directly affect them 5
and others” (p. 24). Is it possible to turn this around, to make schools and universities places where people do ask such questions? How many graduate programs in mathematics education have a class on political aspects of mathematics education? The establishment of such classes might be a good way to start, if change is to occur within our field. REFERENCES Abreu, G. de (2002). Mathematics learning in out-of-school contexts: A cultural psychology perspective. In L. English (Ed.), Handbook of international research in mathematics education: Directions for the 21st century (pp. 323–353). Mahwah, NJ: Erlbaum. Apple, M. W. (2000). Mathematics reform through conservative modernization? Standards, markets, and inequality in education. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 243–259). Westport, CT: Ablex. Boaler, J. (Ed.). (2000). Multiple perspectives on mathematics teaching and learning. Westport, CT: Ablex. Chomsky, N. (2000). Chomsky on miseducation. Lanham, MD: Rowman and Littlefield. Civil, M. (2002). Everyday mathematics, mathematicians’ mathematics, and school mathematics: Can we bring them together? In M. Brenner and J. Moschkovich (Eds.), Journal of Research in Mathematics Education Monograph #11: Everyday and academic mathematics in the classroom (pp. 40–62). Reston, VA: National Council of Teachers of Mathematics. D’Ambrosio, U. (1999). Literacy, matheracy, and technoracy: A trivium for today. Mathematical Thinking and Learning, 1, 131–154. Davis, P. J., & Hersh, R. (1981). The mathematical experience. Brighton, England: Harvester. Davis, P. J., & Hersh, R. (1986). Descartes’ dream: The world according to mathematics. Brighton, England: Harvester. Er-Sheng, Ding (1999). Mathematics curriculum reform facing the new century in China. In Z. Usiskin (Ed.), Developments in mathematics education around the world, Volume 4 (pp. 58–70). Reston, VA: National Council of Teachers of Mathematics. Fasheh, M. (1997). Is math in the classroom neutral—or dead? For the Learning of Mathematics, 17(2), 24–27. Fasheh, M. (2000, September). The trouble with knowledge. Paper presented at the meeting A global dialogue on building learning societies—knowledge, information and human development, Hanover, Germany.
Gutstein, E. (2003). Home buying while brown or black. Rethinking Schools, 18(1), 35–37. Harper’s Index (2003, September). Harper’s Magazine, p. 13. Horn, R. A. Jr., & Kincheloe, J. L. (Eds.), (2001). American standards: Quality education in a complex world: The Texas case. New York: Peter Lang. Julie, C. (1993). People’s mathematics and the applications of mathematics. In J. de Lange, I. Huntley, C. Keitel, & M. Niss (Eds.), Innovating in mathematics education by modeling and applications (pp. 32–40). London: Ellis Horwood. Kilpatrick, J. (1981). The reasonable ineffectiveness of research in mathematics education. For the Learning of Mathematics, 2(2), 22–29. Moll, L. C., & Greenberg, J. B. (1992). Creating zones of possibilities: Combining social contexts for instruction. In L. C. Moll (Ed.), Vygotsky and education: Instructional implications of sociocultural psychology (pp. 319–348). Cambridge: Cambridge University Press. Mukhopadhyay, S. (1998). When Barbie goes to classrooms: Mathematics in creating a social discourse. In C. Keitel (Ed.), Social justice and mathematics education (pp. 150–161). Berlin: Freie Universitat. Mukhopadhyay, S., & Greer, B. (2001). Modeling with purpose: Mathematics as a critical tool. In B. Atweh, H. Forgasz, & B. Nebres (Eds.), Sociocultural Research on mathematics education: An international perspective (pp. 295-312). Mahwah, NJ: Erlbaum. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author. Skovsmose, O. (2000). Aporism and critical mathematics education. For the Learning of Mathematics, 20(1), 2–8. Skovsmose, O., & Valero, P. (2002). Democratic access to powerful mathematical ideas. In L. English (Ed.), Handbook of international research in mathematics education: Directions for the 21st century. Mahwah, NJ: Erlbaum. Slavin, R. E. (2002). Evidence-based education policies: Transforming educational practice and research. Educational Researcher, 31(7), 15–21. Usiskin, Z. (1999). Is there a worldwide mathematics curriculum? In Z. Usiskin (Ed.), Developments in mathematics education around the world, Volume 4 (pp. 213–227). Reston, VA: National Council of Teachers of Mathematics. Vithal, R., & Valero, P. (2001). Researching mathematics education in situations of social and political conflict. Roskilde, Denmark: Centre for Research in Learning Mathematics.
Freire, P., & Shor, I. (1987). A pedagogy for liberation. Westport, CT: Bergin & Garvey.
Volmink, J. D. (1999). School mathematics and outcomes-based education: A view from South Africa. In Z. Usiskin (Ed.), Developments in mathematics education around the world, Volume 4 (pp. 84–95). Reston, VA: National Council of Teachers of Mathematics.
Freudenthal, H. (1978). Weeding and sowing. Dordrecht, The Netherlands: Reidel.
Wilgoren, J. (2001, June 6). Repetition + Rap = Charter School Success. New York Times, pp. A1.
Gándara, P., Rumberger, R., Maxwell-Jolly, J., & Callahan, R. (2003). English learners in California schools: Unequal resources, unequal outcomes. Education Policy Analysis Archives, 11, Number 36.
Winerip, M. (2003, August 13). The ‘zero dropout’ miracle: Alas! Alack! A Texas tall tale. New York Times, p. B7. (Reprinted in Rethinking Schools, 18(1), 8).
Gatto, J. T. (2003). The underground history of American education. New York: Oxford Village Press. 6
What is Mathematics Education For?
The Mathematics Educator 2003, Vol. 13, No. 2, 7â€“21
Hidden Assumptions and Unaddressed Questions in Mathematics for All Rhetoric Danny Bernard Martin In this article, I discuss some of the hidden assumptions and unaddressed questions in the increasingly popular Mathematics for All rhetoric by presenting an alternative, critical view of equity in mathematics education. Conceptualizations of equity within mainstream mathematics education research and policy have, for the most part, been top-down and school-focused in ways that marginalize equity as a topic of inquiry. Bottom-up, community-based notions of education in mathematics education are often of a different sort and more focused on the connections, or lack thereof, between mathematics learning and real opportunities in life. Because of these differences, there has been a continued misalignment of the goals for equity set by mathematics educators and policy makers in comparison to the goals of those who continue to be underserved in mathematics education. I also argue that equity discussions and equity-related efforts in mathematics education need to be connected to discussions of equity in the larger social and structural contexts that impact the lives of underrepresented students. Achieving Mathematics for All in the context of limited opportunity elsewhere may represent a Pyrrhic victory. Portions of this paper are based on the authorâ€™s published dissertation, Martin (2000), postdoctoral work, Martin (1998), and an earlier paper, Martin (2002a), presented at the Annual Meeting of the American Educational Research Association, New Orleans, 2002.
In this article, I discuss some of the hidden assumptions and unaddressed questions in the increasingly popular Mathematics for All rhetoric by presenting an alternative, critical view of equity rhetoric in mathematics education. My arguments will probably generate more questions than answers, but it is my hope that any subsequent discussion serves as a catalyst to move mathematics educators beyond the rhetoric stage in this movement toward meaningful action. Mathematics for All is a worthy philosophical approach to mathematics education. However, mathematics educators should not be satisfied with working toward equity in mathematics education simply for the sake of equity in mathematics education and settling for small victories like Mathematics for All. For reasons of social justice, I also argue that equity discussions and equity-related efforts in mathematics education should extend beyond a myopic Danny Bernard Martin is Professor and Chair of Mathematics at Contra Costa College. He received his Ph.D. in Mathematics Education from the University of California, Berkeley and was a National Academy of Education Spencer Postdoctoral Fellow in 1998-2000. His research focuses on equity issues in mathematics education; mathematics learning among African Americans; and mathematics teaching and curriculum issues in middle school, secondary, and community college contexts. He is also interested in school-community collaborations. Danny Bernard Martin
focus on modifying curricula, classroom environments and school cultures absent any consideration of the social and structural realities faced by marginalized students outside of school and the ways that mathematical opportunities are situated in those larger realities (e.g., Abraham & Bibby, 1988; Anderson, 1990; Apple, 1992/1999, 1995; Campbell, 1989; Dâ€™Ambrosio, 1990; Frankenstein, 1990, 1994; Gutstein, 2002, 2003; Martin, 2000b; Martin, Franco, & Mayfield-Ingram, 2003; Stanic, 1989; Tate, 1995; Tate & Rousseau, 2002). Empty Promises and Prior Reforms In order to add a bit of historical context to my critical remarks, I want to briefly revisit three interrelated events within mathematics education, each occurring about fifteen years ago. An Attempt to Frame Equity and Achievement. The first event occurred in 1988. In that year, Reyes and Stanic (1988) published one of the most significant pieces of literature on issues of race, sex, socioeconomic status, achievement, and persistence to have appeared within the field at that time. In that article, they provided a useful, although incomplete, theoretical framework to explain differences in mathematics achievement. That framework served as a foundation upon which to base future research on 7
equity issues in mathematics education. The paper called for studies exploring relationships among the following factors: teacher attitudes, societal influences, school mathematics curricula, classroom processes, student achievement, student attitudes, and student achievement-related behaviors. Creating Standards for Mathematics Learning. The second event occurred in 1989. The National Council of Teachers of Mathematics (NCTM) published its Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989). The authors of that document were given two charges (a) create a coherent vision of what it means to be mathematically literate both in a world that relies on calculators and computers to carry out mathematical procedures and in a world where mathematics is rapidly growing and is extensively being applied in diverse fields and (b) create a set of standards to guide the revision of the school mathematics curriculum and its associated evaluation toward this vision. In addition to creating a vision for mathematical literacy and setting standards for school mathematics, the Curriculum and Evaluation Standards (NCTM, 1989) also contained very strong statements about equity, stressing the fact that all students should learn mathematics, not just the college-bound or (white) males: 1.
The social injustices of past schooling practices can no longer be tolerated. Current statistics indicate that those who study advanced mathematics are most often white males. Women and most minorities study less mathematics and are seriously underrepresented in careers using science and technology. Creating a just society in which women and various ethnic groups enjoy equal opportunities and equitable treatment is no longer an issue. Mathematics has become a critical filter for employment and full participation in our society. We cannot afford to have the majority of our population mathematically illiterate: Equity has become an economic necessity. (p. x)
Finally, in developing the standards, we considered the content appropriate for all studentsâ€Ś. The mathematical content outlined in the Standards is what we believe all students will need if they are to be productive citizens in the twenty-first century. If all students do not have the opportunity to learn this mathematics, we face the danger of creating an intellectual elite and a polarized society. The image of a society in which a few have the mathematical knowledge needed for the control of economic and scientific development is not consistent either with the values of a just democratic system or with its
economic needs. We believe that all students should have an opportunity to learn the important ideas of mathematics expressed in these standards. (p. x)
These statements constituted the early tenets of the Mathematics for All movement and characterized the early discourse surrounding this movement. Along with similar statements found in other reform-oriented documents (e.g., National Research Council [NRC], 1989), they also alluded to the fact that African American, Latino, Native American, female, and poor students have traditionally trailed their White and Asian American peers on most measures of achievement and persistence and have lacked access to the kind of mathematics that allows them to fully function in school and society (Meyer, 1989; Tate, 1997; Tate & Rousseau, 2002). Further, these statements acknowledged that both policy and curriculum changes are needed to help reverse these trends. Underserving a Generation of Students. At the time, the authors of the Curriculum and Evaluation Standards may have believed that strong statements about equity, in combination with the principles outlined in that document, would lead to the kind of reform efforts that would help alleviate inequities in achievement and persistence for future generations of underserved students. However, achievement and persistence data show that African American, Latino, Native American, and many poor students continued to experience these inequities (e.g., Tate, 1997). By way of evidence for this last statement, consider the third event, which occurred in 1990. It was then that the students representing the Class of 2002 entered first grade. Looking back, I would argue that many of those students were not well-served by more than a decade of mathematics education reform and strong statements about equity. Data from the Third International Mathematics and Science Studies (TIMSS) show that American students, as a group, continued to lag behind their peers in many countries (Schoenfeld, 2002). It can also be argued that the most underserved students of the Class of 2002 were large numbers of African American, Latino American, Native American, and poor students. National Assessment of Educational Progress (NAEP) data over the past fifteen years reveal that although there have been some modest gains in mathematics achievement and persistence by these students (Schoenfeld, 2002), disparities continue to exist and there is evidence that
differences in achievement may be increasing once again (e.g., Lee, 2002). The convergence of affairs described in the three events above leads me to the following conclusion: Despite strong equity-oriented discourse in the 1989 Curriculum and Evaluation Standards, the development of equity-based frameworks such as those outlined by Reyes and Stanic (1988) and others (Oakes, 1990; Secada, 1989, 1992; Secada, Fennema, & Adajian, 1995; Secada, Ogbu, Peterson, Stiff & Tonemah, 1994) and despite increased understandings of how students learn, how teachers teach, and improved methods of assessing teachers and students—math educators have yet to produce adequate solutions to differential achievement and persistence along ethnic lines. Equity in mathematics education remains elusive more than a decade following the three events described above. Renewing the Promise Nearly fifteen years after publication of the Curriculum and Evaluation Standards, the architects of mathematics education reform have produced an updated standards document entitled Principles and Standards for School Mathematics1 (NCTM, 2000). According to Schoenfeld (2002), the Standards are “a vision statement for mathematics education designed to reflect a decade’s experience since the publication of the [Curriculum and Evaluation Standards]” (p. 15). They make explicit the mathematics that is valued and describe the goals for learning this valued mathematics (i.e., mathematics for life, mathematics as a part of cultural heritage, mathematics for the workplace, and mathematics for the scientific and technical community). In the March 2002 issue of the NCTM News Bulletin, NCTM past-President Lee Stiff confirmed this when he stated that the 1989 Curriculum and Evaluation Standards “described the teaching and learning that were valued. In the updated version of this document… the teaching and learning outcomes that we continue to value are revisited” (p. 3). Like their 1989 counterpart, the 2000 Standards also indicate which students should learn this valued mathematics (i.e., Mathematics for All), how they might go about learning it, and how we should assess both teachers and students as they attempt to teach and learn it. Noticeably absent are references to teaching and learning mathematics for social justice; that is, having those who have been traditionally shut out of the mathematics pipeline learn mathematics to help them improve the conditions of their lives.
Danny Bernard Martin
In effect, the old and new Standards documents describe what Apple (1992/1999, 1993) calls the official knowledge of mathematics education. This term is an outgrowth of Apple’s contentions that some forms of knowledge are more valued than others and that these preferred “forms of curricula, teaching, and evaluation in schools are always the results of such accords or compromises where dominant groups, in order to maintain their dominance, must take the concerns of the less powerful into account” (1993, p. 10). Apple indicated that these compromises “are usually not impositions, but signify how dominant groups try to create situations where the compromises that are formed favor them” (p. 10). Having identified mathematics knowledge as a form of high-status knowledge and having invoked the questions of What knowledge is of most worth? and Whose knowledge is of most worth?, Apple offered a critical analysis of the 1989 Curriculum and Evaluation Standards. This critique challenged the notions of mathematics literacy called for in the standards. In one part of that analysis, he stated: The recognition that mathematical knowledge is often produced, accumulated, and used in ways that may not be completely democratic requires us to think carefully about definitions of mathematical literacy with which we now work and which are embedded in Standards volumes…. My arguments in this article are based on a recognition that there is a complex relationship between what comes to be called official knowledge in schools and the unequal relations of power in the larger society…. I have claimed that one of the primary reasons that mathematics knowledge is given high status in current reform efforts is not because of its beauty, internal characteristics, or status as a constitutive form of human knowing, but because of it socioeconomic utility for those who already possess economic capital. In order for our students to see this and to employ mathematics for purposes other than the ways that now largely dominate society, a particular kind of mathematics literacy may be required. (Apple, 1992/1999, p. 97-98)
In light of this critique, it is reasonable to ask whether the updated Standards address the substance of Apple’s concerns. The updated Standards are based on six core principles: equity, curriculum, teaching, learning, assessment, and technology. Because equity is listed first among the core principles, there appears to be an implied promise that Standards-based reform will result in the kind of significant change that will be necessary to improve achievement and persistence among marginalized students (Martin, Franco, et al, 9
2003). Like its 1989 counterpart, the 2000 Standards volume also indicates which students should learn mathematics, how they might go about learning it, and how we should assess both teachers and students as they attempt to teach and learn mathematics. Moreover, the language of Mathematics for All continues to emanate from this and other recent documents that discuss standards (e.g., RAND Mathematics Study Panel, 2003). In writing about standards and equity, Alan Schoenfeld, who is widely recognized as a leader in the field of mathematics education, recently stated “Mathematical literacy should be a goal for all students” (2002, p. 13). Building on the ideas of Robert Moses (Moses, 1994; Moses & Cobb, 2001; Moses, Kamii, & Swap, 1989), Schoenfeld also likened mathematics literacy to a new form of civil rights, highlighting the belief that “the ongoing struggle for citizenship and equality for minority people is now linked to an issue of math and science literacy” (Moses, 1994, p. 107). It is important to accept such statements by leaders in the field as good-faith efforts to bring attention to the inequities faced by marginalized students in mathematics education. However, to ensure that such statements about equity and Mathematics for All do not amount to another decade of empty promises and sloganizing, I believe that continued interrogation, similar to Apple’s (1992/1999) critique of the Curriculum and Evaluation Standards, should be extended to the 2000 Standards and other current reform efforts that claim to have equity as a goal. Only through ongoing critical analysis and reflection is it possible to ensure that attention to the issues affecting mathematics achievement and persistence among African American, Latino, Native American, and poor students remain front and center and that high quality mathematics teaching, learning, curriculum, and life opportunities become a reality for these students, many whom have lacked access to and benefited very little from previous reform efforts, despite strong pronouncements about equity (Martin, Franco, et al, 2003). Indeed, if one compares the discourse about equity found in the 1989 Curriculum and Evaluation Standards (see above) to that found in the 2000 Standards—which is limited to statements about high expectations and strong support for all students—it is very apparent that earlier language stressing mathematics learning for liberatory purposes and having marginalized students use mathematics to critically analyze the conditions in which they live has subsided. In fact, the Equity Principle of the Standards contains no explicit or particular references to African 10
American, Latino, Native American, and poor students or the conditions they face in their lives outside of school, including the inequitable arrangements of mathematical opportunities in these out of school contexts. I would argue that blanket statements about all students signals an uneasiness or unwillingness to grapple with the complexities and particularities of race, minority/marginalized status, differential treatment, underachievement in deference to the assumption that teaching, curriculum, learning, and assessment are all that matter2 (e.g., NRC, 2002). A recent pronouncement by the NRC (2002) involving research on the influence of standards on mathematics and science education held that rigorous research, by definition, cannot be conceptualized as advocacy work. However, the quest to make sure that equity issues are brought to the fore, and remain there, in mathematics education research will involve the kind of advocacy work that some do not see as legitimate. Mathematics for All: How Do We Get There? In addition to the publication of the Principles and Standards for School Mathematics, a potentially influential paper on equity issues in mathematics education has appeared. That paper, authored by Allexsaht-Snider and Hart (2001)3, is entitled “Mathematics for All”: How Do We Get There?. In it, the authors synthesize progress on equity issues in mathematics education over the last decade. Based on their reviews of the research literature and of analysis of math education reform, they suggest three areas of focus for continued research: structural aspects of school districts, teacher beliefs about diverse students and the learning of mathematics, and classroom practices. The paper by Allexsaht-Snider and Hart is especially timely because its appearance, against the backdrop of persistently low achievement by minority and poor students and critiques such as that leveled by Apple (1992/1999), leads to questions about how far mathematics education for under-represented students has evolved and questions about how researchers and policy-makers will respond to a host of complex equity-related issues that were not of paramount importance fifteen years ago: • Rapidly changing demographics that will continually challenge our definitions of equity and diversity, both in terms of defining student populations and determining what resources are needed to help these students excel (e.g., Day, 1993).
Changing curriculum and course-taking policies in many school districts that now require all students, despite their prior preparation, to enroll in algebra by 8th or 9th grade. High-stakes testing in mathematics that will have a disproportionately negative impact on underrepresented students given that many of these students have less access to high-quality teaching and curriculum and that accountability measures for low test performance are often punitive in nature (e.g., Gutstein, 2003; Tate, 1995; Tate & Rousseau, 2002). A changing economy that now relies on large numbers of foreign-born workers to fill math and science-based technical jobs and less on the large pool of under-represented students who remain on the periphery of mathematics and science.
Critiquing Equity and Mathematics for All Rhetoric In my view, an analysis of equity in mathematics education that takes into the account the issues raised above and that contemplates the tensions that these considerations raise for Mathematics for All will help move mathematics educators beyond the rhetoric stage. Below, I attempt such an analysis by focusing on four main themes: (1) the marginalization of equity issues within mathematics education research, (2) the misalignment of top-down and bottom-up approaches to equity, (3) restrictive definitions of equity, and (4) the need to situate equity concerns within a broader conceptual framework that extends beyond classrooms and curricula. Complicity and Marginalization of Equity Issues. Echoing similar claims made by others (e.g., Cobb & Nasir, 2002; Gutstein, 2002, 2003; Khisty, 2002; Secada, 1989; Secada et al, 1995), I suggest that a major reason the mathematics education community has struggled with achievement and persistence issues among underrepresented students, and why effective solutions have been slow in coming, lies in how mathematics educators have dealt with equity-related issues, both in terms of the theoretical frameworks and analytical methods that have been employed and the equity-related goals that have been set. If we examine the way that the “equity problem” in mathematics education has been situated and defined relative to the other research that gets done, it can be said that, contrary to its listing at the first principle in the Standards, equity has been a marginalized topic in mathematics education (Meyer, 1989; Secada, 1989, 1991, 1992; Secada et al, 1995; Skovsmose & Valero, Danny Bernard Martin
2001; Thomas, 2001). Discussions of equity within mathematics education have typically been confined to special sessions at conferences, special issues of journals, or critical issues sections of books. In my view, the status of African American, Latino, Native American, and poor students has not been a primary determinant driving mathematics education reform. When discussions do focus on increasing participation among these students, it is usually in reference to workforce and national economic concerns. Secada (1989) called this “enlightened self-interest.” Gutstein (2003) stated “to discuss equity from the perspective of U.S. economic competition is to diminish its moral imperative and urgency” (p. 38). Even within the context of the “math wars4,” an intense political and philosophical debate between those supporting traditional, skills-focused approaches to mathematics teaching and learning and those supporting approaches called for in the 1989 Curriculum and Evaluation Standards (i.e. a focus on conceptual understanding, connections, mathematical communication, multiple representations, and analyzing data), the needs of marginalized students have never been the center of discussion in these very public arguments. As such arguments have raged on among academics and politically powerful interest groups, marginalized students have continued to suffer low achievement and limited persistence. When a similar debate about skills versus process approaches to writing erupted in the field of literacy, Delpit (1995) had the following to say: In short, the debate is fallacious; the dichotomy is false. The issue is really an illusion created initially not by teachers but by academics whose worldview demands the creation of categorical divisions—not for the purpose of better teaching, but the for the goal of easier analysis. As I have been reminded by many teachers... those who are most skilled at educating black and poor children do not allow themselves to be placed in “skills” or “process” boxes. They understand the need for both approaches. (p. 46)
I would also argue that such debates are symptomatic of a certain kind of complicity that has been largely ignored in discussions involving equity, accountability, and standards-setting. Despite strong statements about equity that were included in the 1989 Curriculum and Evaluation Standards and despite the fact that equity has been listed as the first cornerstone principle of the 2000 NCTM Standards, one of the great paradoxes of mathematics education reform over the last fifteen years is that the very same community 11
that has engineered these reforms also has the dubious distinction of overseeing the inequities in achievement and persistence that have characterized the experiences of many poor and minority students (Martin, Franco, et al, 2003). Because equity concerns have not been central to mainstream mathematics education research, there is also a risk that recent attention to these issues have turned equity into the “problem of the day” in the same way that trends in mathematics education research have shifted from one “theory of the day” to another whether it be cognitive analyses, constructivism, or situated learning. The last several years have seen the rise of cognitive and decontextualized analyses (e.g., Davis, 1986; Schoenfeld, 1985, 1987) followed by a transition to situated analyses (e.g., Anderson, Reder, & Simon, 1996; Brown, Collins, & Duguid, 1989; Cobb, 2000; Cobb & Bowers, 1999; Cobb, Yackel, & Wood, 1992; Lave & Wenger, 1991). The first research approach has resulted in studies that include marginalized students but that do not explicitly address the social and contextual factors that contribute to their underachievement, focusing instead on content and problem-solving behaviors. Studies in the situated approach have addressed issues of context but in such limited ways that discussions of differential socialization, stratification, opportunity structure, ethnicity, and social class are often noticeably absent. Misalignment of Top-Down and Bottom-Up Approaches. Rather than responding directly to the needs of marginalized students and centering discussions around what is best for these students, policy makers and mathematics educators have decided what (valued) mathematics should be learned, who should learn this mathematics, and for what purposes equity in mathematics is to be achieved. I want to suggest that conceptualizations of equity within mathematics education have, for the most part, been top-down and school-focused. Very little equity research and policy has focused on bottom-up, community-based notions of equity (e.g., Moses, 1994; Moses & Cobb, 2001). Class (2002), for example, has stated that such bottomup approaches are unusual among education reformers, who typically focus on curriculum, teaching, and test scores and who believe that equity has been achieved “when differences among sub-groups... of students are disappearing” (Allexsaht-Snider & Hart, 2001, p. 93) as a result of fixing or remedying curriculum, teachers, and funding streams.
On the other hand, equity in mathematics education, as defined by marginalized students, parents, and community members is likely to be related as much to their day-to-day experiences in those outof-school contexts whose participation is mediated or dictated by knowledge of mathematics as it is to their school-based experiences (Anhalt, Allexsaht-Snider, & Civil, 2002; Civil, Andrade, & Anhalt, 2000; Civil, Bernier, & Quintos, 2003; Lubienski, 2003; Martin, 2000, 2003; Perissini, 1997, 1998). In my own research with African American adults and adolescents, I have found that a failure to benefit from mathematics knowledge, both real and perceived, and perceptions about limitations in the larger opportunity structure has an impact on the desire to invest or re-invest in mathematics learning (Martin, 2000, 2003). Because of the differences in these top-down and bottom-up approaches to equity, interventions formulated by mathematics educators have remained, and are likely to remain, out of alignment with the inequities experienced by underrepresented students, parents, and communities. Defining Equity in Mathematics Education. How has equity in mathematics education been defined and what essential elements of these working definitions are missing? I raise this question because if we are to get there, it certainly helps to understand where there is. Moreover, the definitions we use in solving problems also serve as intellectual compasses for the solution routes that we take. As a starting point in their discussion, Allexsaht-Snider and Hart (2001) define5 equity in mathematics education as follows: Equity in mathematics education requires: (a) equitable distribution of resources to schools, students, and teachers, (b) equitable quality of instruction, and (c) equitable outcomes for students. Equity is achieved when differences among sub-groups in these three areas are disappearing. (p. 93)
This definition is in response to the welldocumented disparities in achievement and persistence outcomes that have remained among between African American, Latino, Native American, and poor students on the one hand and many White and Asian American students on the other. This is significant because it is only recently that definitions of equity in mathematics education have addressed the students to whom we now apply them. Past concerns with educating the best and the brightest to achieve national competitiveness for the United States have now shifted to a concern for mathematics for all. That is, definitions of and Hidden Assumptions
approaches to equity in mathematics education have ranged from being highly selective and conditional to being as broad and non-specific as mathematics for all. Rather than centering our discourse in mathematics education on the relationships between mathematics learning and the kind of mathematics that leads to real opportunities in the lives for marginalized students, what I call opportunity mathematics (Martin, 2003; Martin, Franco, et al, 2003), we have continued to norm our efforts and discussions around White, middle-class students and the kinds of mathematics that they have long been given access (Stanic, 1989). I would further argue that too little of the mathematics learned by many African American, Latino, Native American, and poor students leads to the kinds of opportunities that improve their conditions in life. Enrollment patterns in high-status mathematics courses substantiate this claim (e.g., Oakes, 1990; Tate & Rousseau, 2002). If we are truly interested in critically examining issues of equity so that we can be more responsive to the needs of students, teachers, parents, and communities, several questions must be brought to bear: Do our definitions of equity gloss over the deeply embedded structures that produce inequities? Do reform-minded equity efforts get transformed in ways that continue to leave some groups on the outside looking in? Do theoretical perspectives and equityoriented rhetoric take into account the collective histories of the groups for whom equity is desired, resisting the temptation to attribute low achievement to race and ethnicity instead of highlighting the devastating effects of raci s m and the way that schooling and curriculum has contributed to differential opportunities to learn (Apple, 1992/1999). Most important, will we resist the temptation to accept short-term gains (i.e. all students taking algebra) as evidence that equity in mathematics education has been achieved? Rather than restricting our definitions of and goals for equity to equal access, equal opportunity to learn, and equal outcomes, I would like to suggest that math educators working to eliminate inequities seek to extend Allexsaht-Snider and Hart’s (2001) three areas of focus. A focus on structural aspects of school districts, teacher beliefs about diverse students, and classroom practices is important but, in many ways, this focus does not allow us to situate disproportionate achievement and persistence patterns within a broader conceptual framework of sociohistorical, structural, community, school, and intrapersonal factors (Atweh, Forgasz, & Nebres, 2001; Gutstein, 2003; Martin, Danny Bernard Martin
2000; Oakes, 1990). I suggest that a fourth goal of equity research should be to empower students and communities with mathematics knowledge and literacy as a powerful act of working for social justice and addressing issues of unequal power relations among dominant and marginalized groups (e.g., Abraham & Bibby, 1988; Anderson, 1990; Apple, 1992/1999; D’Ambrosio, 1990; Frankenstein, 1990, 1994; Gutstein, 2002, 2003; Moses & Cobb, 2001). Comments by Apple (1992/1999) are helpful in clarifying this broader conceptual framework: Education does not exist in isolation from the larger society. Its means and ends and the daily events of curriculum, teaching, and evaluation in schools are all connected to patterns of differential economic, political, and cultural power…. That is, one must see both inside and outside the school at the same time. And one must have an adequate picture of the ways in which these patterns of differential power operations operate. In a society driven by social tensions and by increasingly larger inequalities, schools will not be immune from—and in fact may participate in recreating—these inequalities. If this is true of education in general, it is equally true of attempts to reform it. Efforts to reform teaching and curricula—especially in such areas as mathematics that have always been sources of social stratification, as well as possible paths of mobility—are also situated within these larger relations. (p. 86)
Situating Equity Within a Broader Conceptual Framework. Some might ask What is the marginal gain in adding this fourth goal? I believe, as do others who support this goal (e.g., Frankenstein, 1990, 1994; Gutstein, 2002, 2003; Ladson-Billings, 1995; Tate, 1995), considerations of social justice force mathematics educators to think beyond curriculum and classrooms so as to situate mathematics learning for marginalized students within the larger contexts that impact their lives. Without attention to the ways in which the arrangement of mathematical, and other, opportunities outside of school further contributes to the marginalization of African American, Latino, Native American, and poor students, I believe equitybased efforts in mathematics education will continue to fall short. Ensuring that marginalized students gain access to quality curriculum and teaching, experience equitable treatment, and achieve at high levels should mark the beginning of equity efforts, not the end. If these students are not able to use mathematics knowledge in liberatory ways to change and improve 13
the conditions of their lives outside of school, they will continue to be marginalized even while mathematics educators and policy makers claim small victories like Mathematics for All. Recent work by Gutstein (2003) with low-income Mexican and Mexican American students and families is also helpful in understanding the goals of a social justice pedagogy in mathematics education. Given the sociopolitical context in which these students and families lived, Gutstein stated “An important principle of a social justice pedagogy is that students themselves are ultimately part of the solution to injustice, both as youth and as they grow into adulthood. To play this role, they need to understand more deeply the conditions of their lives and the sociopolitical dynamics of their world” (p. 39). He set the following goals and objectives for his teaching and his students’ learning: Goals of Teaching for Social Justice
Specific Mathematics-Related Objectives
Develop Sociopolitical Consciousness
Read the World Using Mathematics
Develop Sense of Agency
Develop Mathematical Power
Develop Positive Social/Cultural Identities
Change Dispositions Toward Mathematics
Figure 1: Gutstein’s (2003) goals and objectives for student learning
In one project, entitled Racism in Housing Data?, Gutstein (2003) asked his students to “use mathematics to help answer whether racism has anything to do with the housing prices” (p. 47) in a particular county. More specifically, he asked his students to address questions such as the following: (a) What mathematics would you use to answer that question?, (b) How would you use the mathematics?, and (c) If you would collect any data to answer the question, explain what data you would collect and why you would collect the data. It is clear that Gutstein is attempting to situate mathematics teaching and learning in a context that extends beyond curriculum and classrooms and that he is also attempting to help his students use mathematics to change the conditions of their lives. I also point out that Gutstein’s work and perspective provide evidence for another of my claims: that there are subtle, but important differences, between achieving equity (a goal) and eliminating i n e q u i t y (a process) (Tate, 1995). The first conceptualization—equity as a goal—assumes that there is a point to be reached when all is well and the hard work of getting there can cease. This view also 14
ignores the fact of changing demographics that will continually challenge us to refine our definitions of equity. Although our current conceptions of equity often do not take into account the realities and needs of marginalized groups, new conceptualizations of equity concerns will have to. When those who are marginalized in mathematics begin to exercise their individual and collective agency and power to demand the kind of mathematical literacy leading to real opportunities, policy makers and mathematics educators will have no choice but to listen to these voices and to formulate visions of equity that move these individuals and groups from the periphery of mathematics to the center. The convenient “compromises” described by Apple earlier in this paper will no longer suffice. However, the second conceptualization of equity—as a process—highlights the fact that the necessary hard work will be ongoing and even when gains are made, a high degree of vigilance will be necessary to ensure that needs of marginalized students are attended to and that our definitions of equity are responsive to who these students are, where they come from, and where they want to go in life. In the context of Mathematics for All: How Do We Get There?, mathematics educators may be more focused on achieving the goal of getting there than on the process of how to get there. This is supported by the large number of school districts that now require all students to take algebra in 8th or 9th grade. In the pursuit of this goal, the inequities faced by marginalized students are further compounded because many of them have not been adequately prepared in their earlier mathematical educations due to lack of quality educational resources (e.g., Tate & Rousseau, 2002). Because of a lack of attention to process, the well-meaning goals of Mathematics for All may actually contribute to the inequities faced by underrepresented students. There is a danger that when many of these students do not achieve up to their potential, there will be a tendency to either (a) locate the problem within the student (Boaler, 2002) or (b) assume that contextual forces are so deterministic that students are incapable of invoking agency to resist these forces. Future equity-based research will have to more closely examine how students and contextual forces influence each other. In my view, working to eliminate the inequities faced by marginalized students will require an ongoing commitment that extends beyond simply rendering students eligible for the opportunities that we assume and hope will exist for them. Underrepresented students may experience equal access to mathematics, Hidden Assumptions
have equal learning opportunities, and quantitative data could show equal outcomes. However, these students may still be disempowered if they are not able to use mathematics to alter the power relations and structural barriers that continually work against their progress in life. Let us assume for the moment that the there in Allexsaht-Snider and Hart’s question of Mathematics for All: How Do We Get There? has been reached. The situation is now the following: African American, Latino American, Native American, and poor students now complete substantially more mathematics courses than they did before and their achievement levels have risen to where we deem them acceptable. I pose the same simple, but incisive, questions asked by Gloria Ladson-Billings (2002) during a recent American Educational Research Association symposium: “Now what? What are we going to do for these students?” Will more of these students be allowed to attend the Universities of California at Berkeley and Los Angeles or other universities that are sometimes forced to engage in zero-sum admissions policies (Jones, Yonezawa, Ballesteros, & Mehan, 2002), leaving many qualified minority students on the outside looking in? For example, a state budget crisis has recently forced the Regents of the University of California to consider restricting enrollment at its campuses, signaling a reversal of the state’s commitment to guarantee placement for the state’s top 12% of graduating seniors. That commitment has been in place since 1960. For spring 2003, the university turned away hundreds of mid-year applications from transfer students and freshman. Budget reductions, fee increases, increasing numbers of college-eligible students, and competition for slots have, subsequently, forced many students to the state’s community colleges. The trickle down effect is that many students who have traditionally attended community colleges now find themselves in competition with top-notch high school students. Recently, the state community colleges eliminated 8200 classes, leading to a loss of 90,000 students (Hebel, 2003). Will students who have long used community college as a bridge to higher education now be squeezed out of the community college context and back to their neighborhoods where opportunities are often limited? Will those top high school students now feel that the reward for all their hard work is being taken from them when they are directed to the community college? Where does mathematics fit into all of this? It is well known that mathematics serves as a gatekeeper course for high school graduation and college admissions and many Danny Bernard Martin
students do not gain access to the kind of mathematics to make these graduation and admissions outcomes a reality. Even for those students who are successful in navigating their way to four-year colleges and universities and into math and science majors, it can be asked whether hi-tech companies in Silicon Valley will increase their efforts to recruit these students as engineers and scientists? Will there be an increase in the number of women and minority faculty in mathematics and science departments at colleges and universities? However, such questions may be a case of putting the cart before the horse. If we go back and start with school-mathematics itself, we have to remember that it does not exist in isolation of other curriculum areas. Will marginalized students gain greater access to quality science coursework and instruction? What about literacy? If these students are not able to read and write effectively, how will they be able to handle the rigors of mathematics and science? A common question asked by younger students about mathematics knowledge is How am I going to use this? Convincing students that mathematics learning is worthwhile and can have a significant impact on their lives will be a hard sell for many African American, Latino, Native American, and poor students if they continue to experience inequitable treatment and see few people in their communities who have benefited from mathematics learning or if they are only given access to the kind of mathematics that limits their opportunities in life. I reiterate my earlier point: it is not enough for mathematics educators to work toward equity in mathematics education simply for the sake of equity in mathematics education. Equity discussions and equity-related efforts in mathematics education need to be connected to discussions of equity and in the larger social and structural contexts that impact the lives of underrepresented students. The questions raised above are not intended to throw up a white flag and accept inequities as inevitable. Nor am I suggesting that Mathematics for All is not a worthy goal. However, if achieving equity as a goal in mathematics education means having all students take algebra and, once this is done, that our responsibilities as mathematics educators have been fulfilled, this is, in my view, not an acceptable goal. Mathematics Learning and Literacy in African American Context In advancing my overall arguments, I draw partly from my own research with a diversity of African 15
American adolescents, community college students, parents, and teachers of African American students in two San Francisco Bay Area communities (Martin 2000, 2002b, 2003). For nearly ten years, my ethnographic and participant observation research in these communities has focused on the contextual factors (sociohistorical, structural, community, school, family, peer) that influence well-documented underachievement and limited persistence issues. I have also devoted a great deal of attention to mathematics success and resiliency among adolescents and adults. In particular, I have focused on issues of mathematics socialization and mathematics identity. Mathematics socialization refers to the experiences that individuals and groups have within a variety of mathematical contexts, including school and the workplace, and that legitimize or inhibit meaningful participation in mathematics. Mathematics identity refers to the beliefs that individuals and groups develop about their mathematical abilities, their perceived selfefficacy in mathematical contexts (that is, their beliefs about their ability to perform effectively in mathematical contexts and to use mathematics to solve problems in the contexts that impact their lives), and their motivation to pursue mathematics knowledge. Mathematics socialization and the development of a mathematics identity occur as individuals and groups attempt to negotiate their way into contexts whose participation is mediated or dictated by knowledge of mathematics. Given the wide variety of mathematical practices and contexts in which individuals participate or are denied participation (classrooms, curriculum units, jobs, etc.), mathematics socialization can be conceptualized as both a mechanism for reproducing inequities and for working toward equity in mathematics. A focus on mathematics identity, then, leads to a better understanding of how these experiences operate at a psychological level and give rise to the meanings that people develop about mathematics. I have studied these issues within a broader, multilevel framework that incorporates sociohistorical, community, school, and intrapersonal factors. For the purpose of example, the first two levels of that framework are depicted in Figure 2. I believe that in studying mathematics socialization and mathematics identity issues from a multilevel point of view, I have also gained greater insight into the bottom-up, community-based notions of equity in mathematics education that I mentioned earlier in this paper. Although studies have shown that African American adults and adolescents hold the same folk 16
theories about mathematics as mainstream adults and students, stressing it as an important school subject, few studies have sought to directly examine their beliefs about constraints and opportunities associated with mathematics learning, both for themselves and their children. My research has shown, for example, that African Americansâ€™ conceptions of equity in mathematics education can be deeper, more sophisticated, and even misaligned with those found in reform documents (Martin, 2000, 2002b, 2003). For adults, in particular, I argue that their racialized accounts of their mathematical experiences inside and outside of school reveal that many African American parents situate mathematics learning and the struggle for mathematical literacy/equity within the larger contexts of socioeconomic, political, educational, and African American struggle. As they attempt to become doers of mathematics and advocates for their childrenâ€™s mathematics learning, discriminatory experiences have continued to subjugate some of these parents while others have resisted their continued subjugation based on a belief that mathematics knowledge, beyond its role in schools, can be used to penetrate the larger opportunity structure. I often use case studies (Martin, 2000, 2002b, 2003) to exemplify these varying trajectories of experiences and beliefs about mathematics. Narratives embedded in these case studies often reveal social justice concerns having to do with mathematical opportunity. Sociohistorical Forces Differential treatment in mathematics-related contexts Community Forces Beliefs about African American status and differential treatment in educational and socioeconomic contexts Beliefs about mathematics abilities and motivation to learn mathematics Beliefs about the instrumental importance of mathematics knowledge Relationships with school officials and teachers Math-dependent socioeconomic and educational goals Expectations for children and educational strategies
Figure 2: Mathematics socialization and identity among African Americans: Sociohistorical and community forces.6
In an excerpt from an example that I present elsewhere (Martin, 2000), an African American father offers an insightful opinion about the relationship between African American studentsâ€™ efforts in Hidden Assumptions
mathematics and their perceptions of subsequent opportunity: DM:
Do you think [low motivation is] true for a lot of kids now? Father: I think that’s true for a lot of kids now, yes. DM: It’s mainly that a lot of them don’t see the opportunity attached to [math]? Father: They see the opportunity…. For me, all I wanted was an opportunity. The opportunity wasn’t even there. So, I didn’t pursue it. But what opportunity was there required so much [math] and I satisfied that. Today’s kids, I think, have the opportunity but they need more than just the opportunity. They need the guarantee. DM: Can we guarantee? Father: Yeah, we can. If we will. I mean I can guarantee you that if you do these things, given the way the social structure is set up, there’s a place for you. But you’ve got to set the social structure up first. This view represents just one point in the constellation of African American voices but it offers some evidence for my claim that it will not be enough to achieve equity in mathematics education and settle for that as an end goal. While student ability, teacher bias, tracking, and inadequate curriculum are often cited as causes of low mathematics achievement and limited persistence among African American students (see Martin, 2000), the comments made by this father highlight the fact that not only do adults situate mathematics learning in a larger socioeconomic and political context, but marginalized students may do the same. Addressing teacher bias, tracking, and inadequate curriculum in the name of equity and undoing the role the of mathematics as a gatekeeper may address school-level issues but if students are not able to use mathematics in the out-of-school contexts that define their lives, then underachievement and limited persistence may be rational responses to perceptions of the larger opportunity structure. In addition to my research, my fourteen years of teaching mathematics to students who have often fallen through the cracks and for whom mathematics education reform has done little has convinced me that attempts to achieve equity which focus on content and curriculum issues, teacher beliefs, and school cultures alone will probably have limited impact on negative trends in achievement and persistence if, for example, (1) community forces counteract any good that is done Danny Bernard Martin
within schools despite the best efforts of good teachers who use quality curriculum and exemplary (Standards and non-Standards-based) classroom practices and (2) no attempt is made to leverage these community forces to support in-school efforts designed to eliminate inequity. Eliminating inequities in access, achievement, and persistence in mathematics is not an issue that can be separated from the larger contexts in which schools exist and in which students live. Integrating Theory, Methods, and Practice To improve the status of underrepresented students in mathematics, mathematics educators will need to move beyond the initial rhetoric of Mathematics for All and any tendency to frame equity issues using only the theory and methods of mathematics education. Clearly, our approaches to equity need to be extended in ways that draw on perspectives outside of mathematics education where issues of culture, social context, stratification, and opportunity structure receive greater and more serious attention. Areas like critical social/race theory (e.g., Ladson-Billings & Tate, 1995), sociology of education, and anthropology of education (Ogbu, 1988, 1990) come to mind. Read from one vantage point, one could take from Allexsaht-Snider and Hart’s (2001) definition of equity the assumption that inequities in mathematics education are caused by and can be remedied by fixing school-related factors. Although Allexsaht-Snider and Hart clearly do not assume this, some mathematics educators might. As a result, there might be continued reluctance to analyze the complex social issues that have an impact on mathematics teaching, learning, and disparate outcomes, despite the fact that these issues have been cited in the research literature as being critically important. To return to my preview of recent history in our field, Reyes and Stanic (1988) stated: In the field of mathematics education, there is little, if any, research documentation of the effect of societal influences on other factors in the model. Documenting these connections is both the most difficult and the most necessary direction for future research on differential achievement in mathematics education. (p. 33)
This foregrounding of the complex social issues involved in equity are not yet taking center stage, 15 years later. Finding a way to maintain our concern with mathematics content, mathematics teaching, and learning, while using powerful sociocultural analyses to understand how the arrangement of mathematical opportunities inside and outside of school interact and 17
further contribute to inequities continues to represent the next difficult step in equity-focused research. A second step involves designing meaningful interventions, inside and outside of school, to empower marginalized students with mathematics so that they can change the conditions which contribute to the inequities they face (e.g., Gutstein, 2002, 2003). If equity research in mathematics education is to move forward, we must recognize that inequities in mathematics are reflections of the inequities that exist in out-of-school contexts. Parents, teachers, and students often recognize this parallel to the outside world (e.g., Civil et al, 2000; Civil et al, 2003; Civil & Quintos, 2002; Martin, 2000, 2003; Martin, Franco, et al, 2003) as have critical and progressive mathematics educators (e.g., Abraham & Bibby, 1988; Anderson, 1990; Atweh et al, 2001; Campbell, 1989; D’Ambrosio, 1990; Frankenstein, 1990, 1994; Gutstein, 2003; Hart & Allexsaht-Snider, 1996; Secada & Meyer, 1989; Secada et al, 1995; Tate, 1995). I would also suggest that mathematics educators be wary of transforming equity issues into issues of learning mathematics content. Whether underrepresented students can learn mathematics should not be the main issue of concern. As a field, we should be well beyond deficit-based thinking and trying to fix students so that they conform to normative notions of what a student should be and for what purpose mathematics education should serve these students. Because so much research has been devoted to student failure, there is also the danger that underachievement among underrepresented students will be accepted as the natural and normal starting point for research involving these students. But rich data collected across the many contexts where underrepresented students live and learn will help us reformulate our understanding of both failure a n d success. As a result, we can begin to look for more meaningful explanations and solutions to problematic outcomes and build on what we learn about success. By focusing on diverse contexts, we can begin to uncover a range of solutions focused on what works, where, when, and why, rather than trying to lump all students together and applying one-size-fits-all interventions. Mathematics for All will require that we find a variety of ways to bring underrepresented students into mathematics and a variety of ways—working through schools and communities and at the individual student level—to support their continued development and empowerment.
Conclusion As both a teacher and a researcher, I am a strong advocate of ensuring that all students experience equal access, equal treatment, achieve to their highest potential in mathematics, and participate freely in all forms of mathematical practices that appeal to them inside and outside of schools. I also agree with those who conceptualize mathematics as a gatekeeper and filter (Sells, 1978) and who identify math literacy as a new form of civil right (Moses, 1994; Moses & Cobb, 2001). Yet, I also advocate critical examination of Mathematics for All rhetoric that, in my view, is limited in its vision. By making problematic the there in How Do We Get There?, I hope that my discussion of the hidden assumptions and unaddressed questions in Mathematics for All rhetoric will contribute to a reconceptualization of our equity efforts and our attempts to help students who are marginalized in mathematics. The transition from mathematics for the few to mathematics for all will undoubtedly be an arduous task. As the mathematics education community gives greater attention to equity issues, we cannot assume that Mathematics for All and Algebra for All represent victories over the inequities that marginalized students and their communities face inside and outside of mathematics. Moreover, the people who comprise the communities that we wish to help must become equal partners in mathematics equity discussions and in formulating solutions that address not only content and curricular concerns but issues of social justice as well. It is also my hope that the students who were first graders in the year 2000, the year of the updated Standards, will benefit from a renewed focus and a true desire to move beyond rhetoric so that these students fare better than the Class of 2002. REFERENCES Abraham, J., & Bibby, N. (1988). Mathematics and society. For the Learning of Mathematics, 8(2), 2–11. Allexsaht-Snider, M., & Hart, L. (2001). Mathematics for all: How do we get there? Theory Into Practice, 40(2), 93–101. Anderson, J. R., Reder, L. M., & Simon, H. A. (1996). Situated learning and education. Educational Researcher, 25(4), 5–11. Anderson, S. (1990). Worldmath curriculum: Fighting Eurocentrism in mathematics. Journal of Negro Education, 59(3), 348–359. Anhalt, C., Allexsaht-Snider, M., & Civil, M. (2002). Middle school mathematics classrooms: A place for Latina parents’ involvement. Journal of Latinos and Education, 1(4), 255–262. Apple, M. (1993). Official knowledge: Democratic education in a conservative age. London: Routledge.
Apple, M. (1995). Taking power seriously. In Secada, W., Fennema, E. & Adajian, L. B. (Eds.). New directions for equity in mathematics education (pp. 329–348). Cambridge: Cambridge University Press. Apple, M. (1999). Do the Standards go far enough? Power, policy, and practice in mathematics education. Reproduced in Power, meaning, and identity: Essays in critical educational studies. New York: Peter Lang (Original work published 1992).
Day, J. (1993). Population projections of the United States by age, sex, race, and Hispanic origin: 1993 to 2050. Washington, D.C.: U.S. Department of Commerce (vii–xxiii). Delpit, L. (1995). Other people’s children: Cultural conflict in the classroom. New York: The New Press. Frankenstein, M. (1990). Incorporating race, gender, and class issues into a critical mathematical literacy curriculum. Journal of Negro Education, 59(3), 336–347.
Atweh, B., Forgasz, H., & Nebres, B. (2001). Sociocultural research on mathematics education: An international perspective. Mahwah, NJ: Erlbaum.
Frankenstein, M. (1994). Understanding the politics of mathematical knowledge as an integral part of becoming critically numerate. Radical Statistics, 56, 22–40.
Boaler, J. (2002, April). So girls don’t really understand mathematics?: Shifting the analytic lens in equity research. Paper presented at the annual meeting of the American Educational Research Association, New Orleans, LA.
Gutstein, E. (2002, April). Roads to equity in mathematics education. Paper presented at the annual meeting of the American Educational Research Association, New Orleans, LA.
Brown, J.S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1), 32–42.
Gutstein, E. (2003). Teaching and learning mathematics for social justice in an urban, Latino school. Journal for Research in Mathematics Education, 34(1), 37–73.
Campbell, P. (1989). So what do we do with the poor, non-white female?: Issues of gender, race, and social class in mathematics and equity. Peabody Journal of Education, 66(2), 96–112.
Hart, L., & Allexsaht-Snider, M. (1996). Sociocultural and motivational contexts of mathematics learning for diverse students. In M. Carr (Ed.), Motivation in mathematics (pp. 1–24). Creskill, NJ: Hampton Press.
Civil, M., Andrade, R., & Anhalt, C. (2000). Parents as learners of mathematics: A different look at parental involvement. In M. L. Fernández (Ed.), Proceedings of the Twenty-Second Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 2 (pp. 421–426). Columbus, OH: ERIC Clearinghouse.
Hebel, S. (2003, October 10). California’s budget woes lead colleges to limit access. The Chronicle of Higher Education, p. 21.
Civil, M., Bernier, E., & Quintos, B. (2003, April). Parental involvement in mathematics: A focus on parents’ voices. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago, IL. Civil, M., & Quintos, B. (2002, April). Uncovering mothers’ perceptions about the teaching and learning of mathematics. Paper presented at the annual meeting of American Educational Research Association, New Orleans, LA. Class, J. (2002). The Moses factor [Electronic version]. Mother Jones, May/June. Retrieved November 14, 2003, from http://www.motherjones.com/news/feature/2002/05/ moses.html Cobb, P. (2000). The importance of a situated view of learning to the design of research and instruction. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 45–82). Westport: Ablex. Cobb, P., & Bowers, J (1999). Cognitive and situated perspectives in theory and practice. Educational Researcher, 28(2), 4–15. Cobb, P., & Nasir, N. (Eds.). (2002). Diversity, equity, and mathematics learning [Special double issue]. Mathematical Thinking and Learning, 4(2/3). Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education, Journal for Research in Mathematics Education, 23(1), 2–33. D’Ambrosio, U. (1990). The role of mathematics in building a democratic and just society. For the Learning of Mathematics, 10(3), 20–23.
Jones, M., Yonezawa, S., Ballesteros, E., & Mehan, H. (2002). Shaping pathways to higher education. Educational Researcher, 31(2), 3–12. Khisty, L. (2002, April). Equity in mathematics education revisited: Issues of “getting there” for Latino second language learners. Paper presented at the annual meeting of the American Educational Research Association, New Orleans, LA Ladson-Billings, G. (1995). Making mathematics meaningful in multicultural contexts. In W. G. Secada, E. Fennema, & L. B. Adajian (Eds.), New Directions for Equity in Mathematics Education (pp. 279–297). Cambridge: Cambridge University Press. Ladson-Billings, G. (2002, April). Urban education and marginalized youth in an age of high-stakes testing: Progressive responses. Symposium presented at the annual meeting of the American Educational Research Association, New Orleans. LA. Ladson-Billings, G. & Tate, W.F. (1995). Toward a critical race theory of education. Teachers College Record, 97, 47–68. Lave, J. & Wegner, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press. Lee, J. (2002). Racial and ethnic achievement gap trends: Reversing the progress toward equity? Educational Researcher, 31(1), 3–12. Lubienski, S. (2003, April). Traditional or standards based mathematics?: Parents’ and students’ choices in one district. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago, IL.
Davis, R. (1986). The convergence of cognitive science and mathematics education. Journal of Mathematical Behavior, 5, 321–335. Danny Bernard Martin
Martin, D. B. (1998). Mathematics socialization and identity among African Americans: A multilevel analysis of community forces, school forces, and individual agency. Unpublished postdoctoral project completed for National Academy of Education/Spencer Postdoctoral Fellows program.
Ogbu, J. U. (1988). Diversity and equity in public education: Community forces and minority school adjustment and performance. In R. Haskins & D. McRae (Eds.), Policies for America’s public schools: Teachers, equity, and indicators (pp. 127–170). Norwood: Ablex.
Martin, D. B. (2000). Mathematics success and failure among African American youth: The roles of sociohistorical context, community forces, school influence, and individual agency. Mahwah, NJ: Erlbaum..
Ogbu, J. U. (1990). Cultural model, identity, and literacy. In J. W. Stigler, R. A. Shweder, & G. Herdt (Eds.), Cultural psychology (pp. 520–541). Cambridge: Cambridge University Press.
Martin, D. B. (2002a, April). Is there a there?: Avoiding equity traps in mathematics education and some additional considerations in mathematics achievement and persistence among underrepresented students. Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA.
Perissini, D. (1997). Parental involvement in the reform of mathematics education. The Mathematics Teacher, 90(6), 421–427.
Martin, D. (2002b, April). Situating self, situating mathematics: Issues of identity and agency among African American adults and adolescents. Paper presented at the annual conference of the National Council of Teachers of Mathematics, Las Vegas, NV. Martin, D. (2003, April). Gatekeepers and guardians: African American parents’ responses to mathematics and mathematics education reform. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago, IL. Martin, D., Franco, J., & Mayfield-Ingram, K. (2003). Mathematics education, opportunity, and social justice: Advocating for equity and diversity within the context of standards-based reform. Research brief (draft) prepared for Research Advisory Committee, National Council of Teachers of Mathematics Catalyst conference. Meyer, M.R. (1989). Equity: The missing element in recent agendas for mathematics education. Peabody Journal of Education, 66(2), 6–21. Moses, R. P. (1994). Remarks on the struggle for citizenship and math/science literacy. Journal of Mathematical Behavior, 13, 107–111. Moses, R. P. & Cobb, C.E. (2001). Radical equations: Math literacy and civil rights. Boston MA: Beacon Press. Moses, R., Kamii, M., Swap, S., & Howard, J. (1989). The Algebra Project: Organizing in the spirit of Ella. Harvard Educational Review, 59(4), 423–443. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. National Research Council. (1989). Everybody counts: A report to the nation on the future of mathematics education. Washington, DC: Author. National Research Council. (2002). Investigating the influence of standards: A framework for research in mathematics, science, and technology education. Washington, D.C.: National Academy Press. Oakes, J. (1990). Opportunities, achievement and choice: Women and minority students in science and mathematics. In C. B. Cazden (Ed.), Review of Research in Education, Vol. 16 (pp. 153–222). Washington, DC: AERA.
Perissini, D. (1998). The portrayal of parents in the school mathematics reform literature: Locating the context for parental involvement. Journal for Research in Mathematics Education, 29(5), 555–582. RAND Mathematics Study Panel. (2003). Mathematical proficiency for all students: Toward a strategic research and development program in mathematics education. Washington, DC: RAND. Reyes, L. H., & Stanic, G. (1988). Race, sex, socioeconomic status, and mathematics. Journal for Research in Mathematics Education, 19(1), 26–43. Schoenfeld, A. H. (1985). Mathematical problem solving. New York, NY: Academic Press. Schoenfeld, A. H. (Ed.). (1987). Cognitive science and mathematics education. Hillsdale, NJ: Erlbaum. Schoenfeld, A. (2002). Making mathematics work for all children: Issues of standards, testing, and equity. Educational Researcher, 31(1), 13–25. Secada, W. (1989). Agenda setting, enlightened self-interest, and equity in mathematics education. Peabody Journal of Education, 66(2), 22–56. Secada, W. (1991). Diversity, equity, and cognitivist research. In Fennema, E., Carpenter, T. P., & Lamon, S. (Eds.), Integrating research on teaching and learning mathematics (pp. 17–54). SUNY: Albany. Secada, W. (1992). Race, ethnicity, social class, language and achievement in mathematics. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 623–660). New York: Macmillan. Secada, W., Fennema, E., & Adajian, L. B. (1995). New directions for equity in mathematics education. Cambridge: Cambridge University Press. Secada, W., & Meyer, M. (1989). Needed: An agenda for equity in mathematics education. Peabody Journal of Education, 66(1), 1–5. Secada, W., Ogbu, J. U., Peterson, P., Stiff, L. M., Tonemah, S. (1994). At the intersection of school mathematics and student diversity: A challenge to research and reform. Washington, DC: Mathematical Sciences Education Board and National Academy of Education. Sells, L. W. (1978). Mathematics: A critical filter. Science Teacher, 45, 28–29. Skovsmose, O., & Valero, P. (2001). The critical engagement of mathematics education within democracy. In Atweh, B., Forgasz, H., & Nebres, B. (Eds.), Sociocultural research on mathematics education (pp. 37–56). Hillsdale, NJ: Erlbaum.
Stanic, G. M. A. (1989). Social inequality, cultural discontinuity, and equity in school mathematics. Peabody Journal of Education, 66(2), 57–71. Stiff, L. V. (2002). Reclaiming standards. NTCM News Bulletin, 38(7), 3–7. Tate, W. F. (1995). Economics, equity, and the national mathematics assessment: Are we creating a national toll road? In W. Secada, E. Fennema, & L. Byrd Adajian (Eds.), New directions for equity in mathematics in mathematics education (pp. 191–206). Cambridge: Cambridge University Press. Tate, W. (1997). Race, ethnicity, SES, gender, and language proficiency trends in mathematics achievement: An update. Journal for Research in Mathematics Education, 28(6), 652–680. Tate, W. F., & Rousseau, C. (2002). Access and opportunity: The political and social context of mathematics education. In L. English (Eds.), International Handbook of Research in Mathematics Education (pp. 271–300). Mahwah, NJ: Erlbaum. Thomas, J. (2001). Globalization and the politics of mathematics education. In Atweh, B., Forgasz, H., & Nebres, B. (Eds.) Sociocultural research on mathematics education (pp. 95–112). Hillsdale, NJ: Erlbaum. Weinstein, R. S. (1996). High standards in a tracked system of schooling: For which students and with what educational supports? Educational Researcher, 25(8), 16–19.
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Wilson, S. (2002). California dreaming: Reforming mathematics education. New Haven, CT: Yale University Press.
1 I will refer to this document as the Standards when a distinction against another standards-based document is unnecessary. Occasionally I will collectively refer to both the 1989 and 2000 NCTM documents as the Standards. I have taken care that in each place, the reader will know to which document I refer. 2 These ideas are from discussions that took place in a Working Group on the Changing Nature of Schooling and Demographics led by William Tate and Pauline Lipman at the National Council of Teachers of Mathematics Catalyst Conference held in Reston, VA, September 11-13, 2003. 3 Hart (2001) and Reyes (1988) are the same person. 4
For an account tracing the history of these debates, see Wilson (2002). 5
The prevailing notion is that equity in mathematics is three-pronged: equal access, equal opportunity to learn, and equal outcomes. 6
Readers are urged to see Martin (2000) for a more detailed description of this framework.
MAA-AMS Joint Meeting of the Mathematical Association of America and the American Mathematical Society http://www.ams.org/amsmtgs/2078_intro.html
AMTE Association of Mathematics Teacher Educators http://www.amte.net
San Diego, CA
RCML Research Council on Mathematics Learning http://www.unlv.edu/RCML
Oklahoma City, OK
AERA American Education Research Association http://www.aera.net
San Diego, CA
Mα The Mathematical Association http://m-a.org.uk
NCTM National Council of Teachers of Mathematics http://www.nctm.org
CMESG/GCEDM Canadian Mathematics Education Study Group http://plato.acadiau.ca/courses/educ/reid/cmesg/cmesg.html
Université Laval, Québec, Canada
ICME-10 The 10th International Congress on Mathematics Education http://www.icme-10.dk
HPM History & Pedagogy of Mathematics Conference http://www-conference.slu.se/hpm/about/
PME-28 International Group for the Psychology of Mathematics Education http://igpme.org
JSM of the ASA Joint Statistical Meetings of the American Statistical Association http://www.amstat.org/meetings
PME-NA North American Chapter of the International Group for the Psychology of Mathematics Education http://www.pmena.org
SSMA School Science and Mathematics Association http://www.ssma.org
College Park, GA
AAMT 2005 Australian Association of Mathematics Teachers http://www.aamt.edu.au/mmv
Jan. 17-20 2005
The Mathematics Educator 2003, Vol. 13, No. 2, 23–31
The Fourth “R”: Reflection Norene Vail Lowery Research promotes reflective teaching as an important distinguishing strategy between experienced and novice teachers and is a critical tool for developing teacher knowledge. Reflective teaching practices are supported by national reform efforts and have the potential to affect student achievement in the mathematics classroom. Unfortunately, reflective teaching practices are not always a component of teacher preparation and professional experiences. This discussion highlights: (1) research concerning the importance of teacher reflection; (2) the results of a study implementing reflective teaching practices in an elementary mathematics and science methods course; (3) the resulting “best practices” applied to other teacher learning contexts; and, (4) the benefits of the fourth “R”.
National standards, having emerged from educational reform, promote learning environments that encourage meaningful learning, rather than rote learning, and create a different view of teaching and learning. A component of these reform efforts is to develop teachers who are reflective about teaching and learning. Reflection is seen as what a teacher does when he or she looks back at the teaching and learning that has been experienced, and recreates the events, emotions, and happenings of the situation (Wilson, Shulman, & Richert, 1987). Hoberman and Mailick (1994) believe that learning and competence are gained by practice in performance that involves reflection before, during, and after the action. Research indicates that teacher reflection is a key aspect for obtaining teacher knowledge and pedagogical content knowledge. There exists a stage in which teachers look back on the teaching and learning that has occurred as a means of making sense of their actions and learning from their experiences (Wilson, Shulman, & Richert, 1987). Reflection is seen as a process of reconstructing classroom enactments, including both cognitive and affective dimensions that involve a developmental progression through stages. Experienced and novice teachers differ in their ability to learn from reflection on experience. Reflective experts are more discriminating in their perception and more resourceful in their actions and problem solving. Experienced teachers have a highly developed knowledge base concerning students; notice different Norene Vail Lowery, Ph.D., is an Assistant Professor of Mathematics Education in the Curriculum & Instruction Department of the College of Education at the University of Houston, Houston, Texas. Research interests include elementary and middle school mathematics education, preservice and inservice teacher education, assessment, and the integration of literature and mathematics. Her email address is email@example.com.
Norene Vail Lowery
classroom aspects; are more selective in their use of information during planning and teaching; and, make greater use of instructional and management routines (Borko & Livingston 1989; Borko & Shavelson 1990; Carter, Cushing, Sabers, Stein, & Berliner, 1988). The basis for instructional decisions (teacher’s practical knowledge) is dynamic as it builds through reflective experience (Elbaz, 1983). The Principles and Standards for School M a t h e m a t i c s (National Council of Teachers of Mathematics [NCTM], 2000) advocates new ways of teaching and learning mathematics in the classroom. As mathematics educators begin to implement these guidelines, it is even more crucial that teachers become reflective in practice. However, reflection on teaching is not a traditional component of mathematics instruction. As it is believed that teachers tend to teach the way they were taught, reflective teaching will only be implemented and flourish if teachers become knowledgeable of and are supported by “best practices” in the classroom. The importance of reflective teaching is a central component for designing teaching and learning experiences for teachers. I have implemented a variety of strategies for encouraging reflection into all my courses with both preservice and inservice teachers. Effective protocol for becoming a reflective teacher has emerged through these experiences. This discussion describes original research findings with preservice teachers that led to creating and implementing reflective practices into other teacher education courses. A collective synthesis of these successful efforts resulting in “best practices” is presented for creating learning experiences for teachers that are conducive to begin and support reflective teaching. No longer can there be traditional reliance on just “Reading, ‘Riting, and ‘Rithmetic” as the basic ingredients for providing a quality education. It is vital 23
for all teachers, especially teachers of mathematics, to take a serious look at a fourth “R”: Reflection. Reflective Research Even though the role of reflection in teaching is considered important, reflective action in preservice and inservice teachers is either inhibited by isolation of teachers or by structure of courses and schools (Feiman-Nemser & Buchmann, 1985; Lortie, 1975; Nisbett & Wilson, 1977). Fenstermacher (1994) has suggested reform needed and wanted in teacher preparation requires a tremendous effort in understanding teaching and teacher learning. Other advocates in the renovation of teacher preparation programs appear to confirm this and suggest that such new programs must understand the conditions that promote reflection in beginning teachers. Research findings confirm that the likelihood of long-term success for many novice teachers is hindered by the absence of expert guidance, support, and opportunities to reflect (Tisher, 1978; Veenman, 1984). Examination of the role of reflection involved in learning how to teach can make significant contributions in strengthening the preparation of teachers by complementing a growing knowledge base for teaching. The purpose of this inquiry was to examine the construction of teacher knowledge in learning to teach elementary mathematics and science. Implementing teacher reflection was an integral component of the study. A qualitative methodology was employed and facilitated the discovery of the importance of reflection in learning to teach by preservice teachers in a school-based setting. The respondents were twenty-one junior and senior level interdisciplinary studies majors at a large university who were enrolled in methods courses required for the fulfillment of certification in elementary mathematics and science instruction. The site for this contentfocused (mathematics and science only) professional development school was located on a middle-class, suburban, public K-5 elementary school campus in a central Texas school district. The methods course experience was non-traditional and innovative in: the approach (constructivist1); the content (standardsbased2); the site (school-based with immediate access to inservice teachers and elementary students); and the instructional strategies (reflective practices). Additionally, during the semester, the preservice teachers participated in teaching and tutoring elementary student’s mathematics and science lessons. Groups of three or four preservice teachers were assigned to elementary grade level teachers for creating 24
and teaching these elementary mathematics and science lessons. They debriefed and reflected in small groups, and then individually responded to the course tasks of analyzing, evaluating, and synthesizing all experiences. These activities created the opportunity for the required reflective journals, lab entries, classroom tasks, and summative portfolios that were assigned course products, and consequently used as data sources. In addition, I conducted interviews with the students to obtain additional information. The data were processed following the suggested steps of synthesis of a constant comparative method adapted from Glaser and Strauss (1967). Both evidence and extent of the value in the use reflection were apparent. Knowledge of self, of the learner, and of the task (content) were prevalent themes that emerged from the data. Development and use of the reflective process was an objective of the methods course. Course instructors considered reflective thinking before, during, and after teaching imperative for a thorough teaching experience. The reflective process is valued in professional growth and successful teaching (Dewey, 1933; Schön, 1987). The entire collaborative learning environment was subjected to the reflective process. Reflection on learning occurred in the large group tasks and activities; while working and planning with teachers; while teaching children mathematics and science; while working and planning within grade level groups; and while working and interacting with course instructors. To aid in simplifying the communication of data sources for the purposes of this paper, I will rely on the codes (see Table 1) I used during analysis while I discuss these findings below. Through reflection, much was revealed about the learning components of teacher knowledge and pedagogical content knowledge in mathematics and science. Reflection is not an easy task. Initial reflections in notebooks and in weekly evaluations were superficial and more descriptive in nature rather than reflective (RJ: 12.17.96). Reflections grew in depth and quality over the semester to culminate in the product of the personal portfolio. The portfolio was the ultimate expression of the acquisition of learning that the preservice teachers experienced in this context. The summative portfolio was a deep, elaborate reflection that revealed construction of teacher knowledge. Individual interviews, focus group interviews, and other data sources confirmed and verified results reported in portfolios. These results can largely be
Table 1. Codes developed to distinguish data sources. Source Origin Preservice Teachers (PST)
Data Source Individual interview Overview Focus group interview Overview Lab notebook Overview Weekly Evaluations: Question #: Date Final Examination: Question #: Response # Portfolio: Artifact #: Response #
Field Notes (Researcher) Overview Reflexive Journal (Researcher)
sorted into 4 themes for the purpose of this report: (1) knowledge about self, autonomy, self-efficacy; (2) the importance of confidence and competence; (3) the importance of the value of content, coordinating lesson planning, and questioning; and (4) expanding knowledge about children, assessment, relevancy and group dynamics Knowledge about self. Personal growth was revealed through reflective processes, such as the portfolio. Enthusiasm and patience were among those reported. Knowledge of self as a learner and as a teacher was important. At the end of the semester, the preservice teachers expressed the importance of the reflective process in tracking their growth and experiences. Watching myself evolve into an educator who desires to be a constant learner has been one of the most important changes that have taken place during the semester (FE: Q4: 25).
Confidence and competence. Growth in confidence and competence were reported in the portfolios, the final exams, in weekly evaluations, and through interviews. The preservice teachers were continually encouraged to develop reflective thought as a tool for developing confidence and competence in teaching mathematics and science. ... my attitude toward teaching math and science. Now I know that the best way to teach these subjects is through discovery and exploration, connecting math and science to the real world is vital. I feel more confident about math and science as I enter student teaching (FE: Q4: 6). This experience has been beneficial to my confidence as a teacher and a person in general. ... I have learned from my instructors that science and math must move away from worksheets and become hands-on, minds-on activities. The Norene Vail Lowery
Code PSTI PSTIO PSTF PSTFO LN LNO WE: Q3: 9.4.96 (sample) FE: Q4: 6 (sample) PF: 6: 1 (sample) FN FNO RJ
children must be actively engaged in their learning. Children grow so much deeper when they have opportunities to discover for themselves (PF: 6: 1).
The summative evaluation task, a personal portfolio, required the preservice teachers to confront their own learning and the extent of that learning. Reflecting on previously recorded reflections over the semester was part of the process in the creation of portfolios. I also feel that I have gained confidence in my knowledge of mathematics and science. I always enjoyed these subjects, but did not feel confident about my knowledge until now. This was such a valuable experience because I will carry this confidence with me during my teaching career (FE: Q4: 19). I have always been afraid of math and science. I have never been good at either one because they were boring and abstract to me. Through this semester, I have learned ways to make math and science relevant, fun, and interesting. I now enjoy learning scientific things and events and look forward to teaching them (FE: Q4: 13).
Expanding knowledge of the content. Mathematics and science content learning was experienced as a group and individually. Many preservice teachers had previously had limited or no meaningful experiences with mathematics and science content. There were “Aha’s” while working as learners in content activities. I really enjoyed the math we experienced today. I love math and I am excited to hear that it is not being taught the way I was taught (PSTWE: Q1: 4.9.1). I learned how to explain math in a meaningful way (PSTWE: Q3: 12.14.3).
The preservice teachers had progressed from the lower levels of ability such as question-response 25
techniques, to a level of higher-order thinking skills. From there, they began asking open-ended questions to stimulate children’s thinking and responses. Through reflection on their own deeper learning of the content, teachers identified the value of effective questioning as an important instructional strategy. ... go further than the teacher asking simple questions – have children ask questions and find their own answers (PSTWE: Q2: 3.9.2). Today we worked with individual children who were having trouble with borrowing and subtraction. [The student] didn’t really understand when to borrow and when not to borrow. I picked up that he wasn’t understanding the concept with the way his teacher was explaining it to him. So I took a different approach and I could see that he truly understood the concept after I explained it. At this point, I could tell his confidence was boosted and he had more enthusiasm. Seeing his excitement made me feel great. I had actually taught him something. When we finished with the activity, he asked the teacher if he could work with me some more. I felt really special when he said that (PF: 6.4)!!!
Expanding knowledge of pedagogy. During individual interviews, preservice teachers were asked what they saw as the most important aspects about science/mathematics teaching that an elementary teacher should know. Responses suggested that teachers should work with the students using hands-on, real-world situations; make lessons relevant and challenging; realize that abstract concepts are hard; have more problem solving; know the students; act as a facilitator of learning; and integrate math and science, as well as other areas. Through reflection preservice teachers came to terms with their own attitudes toward mathematics and science, as was evidenced in the data. I also feel that I have gained confidence in my knowledge of mathematics and science. I always enjoyed these subjects, but did not feel confident about my knowledge until now. This was such a valuable experience because I will carry this confidence with me during my teaching career. (FE: Q4: 19). Now I know that the best way to teach these subjects is through discovery and exploration, connecting math and science to the real world is vital. I feel more confident about math and science as I enter student teaching. (FE: Q4.6).
reflective journals, indicated that these preservice teachers attributed the following types of learning to interacting with mathematics and science lessons: content knowledge; the importance of hands-on manipulatives; relevancy; instructional strategies; group dynamics; and student cognitive attributes, abilities and levels. Instructional strategies were identified and implemented, including effective questioning, timing, lesson planning, classroom management, preparation, and authentic assessment. This collaborative interaction was a relevant, authentic learning environment for preservice teachers. Through reflection, preservice teachers were able to track, evaluate, and project their learning. The researcher in this study also used other modes of reflection. A reflexive journal was maintained to record the researcher’s learning, decision-making processes for data collection, analysis, in report writing, and in the embellishments of the field notes, interviews, and others. In this study, reflection was a binding thread for all the experiences of the preservice teachers. Reflective practices allowed the professors and preservice teachers to actively assess, evaluate, and modify the learning experiences. Data indicate that through reflective practices, preservice teachers had a greater sense of self, autonomy, self-efficacy, confidence and competence in teaching mathematics and science, and had incorporated the value of reflection into their belief system. These research findings promoted the implementation of reflective practices used here into other teacher education courses. Inservice teachers have provided similar reports in graduate classes of valuing the use of reflective teaching. From this study many strategies emerged that were effectively used to promote reflection. Best Practices Over time working with these emerging “best practices”, I’ve seen that a belief in the importance of reflection and strategies for becoming a reflective mathematics teacher are developed and progress through three levels: understanding reflection; implementing reflective practices; and developing a reflective venue. In this section, I will share the strategies used and developed in situations such as those in the above research, to illustrate ways to utilize this three-level plan to promote reflective teaching.
Preservice teachers confronted their own learning and the extent of that learning through their summative portfolio assessment. The portfolios, along with the 26
Level One: Understanding the Importance of Reflective Thinking Understanding prior knowledge and beliefs provides a foundation for becoming a reflective teacher. Teachers are encouraged to explore the meaning of “reflection” by actually negotiating an operational definition individually, in small groups, and in whole class discussion. Prompts for Defining Reflection • • •
Write your own definition of “reflection”. What do you perceive as “reflective teaching”? In your group, discuss your responses.
Once common ground is established teachers are asked to put reflection into action. Reflecting on past experiences, prior knowledge, and expectations helps to provide more insight into a teacher’s current perspective on teaching and learning mathematics. Teachers are asked to write an autobiographical sketch that visualizes their own perspectives as a learner and as a teacher. Reflecting on Mathematical Experiences •
What were your experiences with mathematics at school/home/other? What kinds of instructional activities and practices did you experience? Describe any influential teachers, either positive or negative. What are your preferred learning strategies? What do you do to learn? What do you enjoy learning? How did you come to be like this? What stories reveal your roots as a learner? What does this mean for your current instructional strategies? What is (are) your teaching styles/instructional strategies? How might you do things differently? How do you envision mathematics learning for my students?
To validate these reflections, teachers share their thinking in a safe, collaborative group situation only to uncover common attitudes, apprehensions, and beliefs about mathematics. Large group discussion offers an even greater opportunity to connect with peers and to deepen reflective thinking. Many preservice and some inservice teachers are not excited about teaching mathematics. A teacher’s perception of the nature of mathematics greatly influences the mathematics instruction and learning environment in the classroom (Cooney, 1985; Hersch, 1986). The following exercise challenges teachers to come to terms with their own perspectives of Norene Vail Lowery
mathematics teaching and learning. Charged with the task of developing a statement of the nature of mathematics, teachers are encouraged with these writing prompts. Developing a Perspective of the Nature of Mathematics What is mathematics? What is the nature of mathematics? • What are the components of mathematics? • What is the conception of mathematics that you believe is important for your students to know? • What mathematics do you want your students to learn? •
The resulting statements are shared in small groups and then in the large classroom group. This initial draft is revisited periodically to modify and enhance. At the end of the semester, each teacher reviews the first draft, revises, re-writes, or edits as they deem appropriate to complete a final draft to submit. Along with this final draft, teachers submit a “reflective rationale” for any changes that were made in the process. Teachers explain and justify any changes. This allows for growth and clarity of reflective thought and teaching. Teachers need time to “reflect” on the value, complexity, and beauty of mathematics and teaching mathematics. Level Two: Implementing Reflective Strategies – The Reflective Cycle The revelations from Level One create a foundation for implementing the power of reflection into practical classroom applications. Experiences in my undergraduate and graduate classes are created for both field-based preservice and inservice teachers to put reflection into active classroom interaction. These exercises, presented and elaborated below, have successfully enabled teachers to begin reflective thinking and teaching practices. They form a cycle of reflective activity. This reflective cycle includes reflective planning, reflective teaching, and creating a guided reflection. Teachers use prompts individually and then in small groups for self-questioning. Teachers write responses to establish a thinking routine. Small group discourse is used to facilitate reflective thinking practices. As more experience in reflective thinking and teaching develops, many of these become automatic, and the cycle begins to feed upon itself. Reflective planning. A strong emphasis is placed on the first phase of this strategy, reflective planning. Efforts involved in this area appear to benefit all areas of reflection. Teachers are asked to carefully and 27
thoughtfully respond in depth to three components of planning (the lesson, the learner, and the teacher). Reflective Planning The Lesson • What is the content or topic and what prior reasoning, strategies, and thinking skills are expected? • What learning venues are required for learner success? (e.g., communication and representations) • What are expected outcomes? The Learner • What are the goals for the learner? • What are some of the possible misconceptions? • How will the learner be assessed? The Teacher • What must be done to plan and organize? • What are the most effective strategies and questions? (Write some examples.) • What are expected outcomes? What modifications are anticipated? Teachers proclaim this initial step in reflective teaching prepares them for teaching a better lesson by creating competence and confidence. Reflective teaching. Reactive reflection or reflection in action is sometimes quite difficult if not impossible to recall after the events. However, reflection here is crucial. Experienced teachers have a rich repertoire of exemplars from which to choose when confronted with problematic situations in the classroom. Experienced teachers may demonstrate an uninterrupted flow of teaching easily adapting to the unexpected. As more classroom experience is gained, novice teachers develop their own similar resources. Why was this done? What questions prompted the course of the teaching and learning? Making mental notes may be all that is possible to do. However, written notes are very effective. Periodic video or/and audio taping of teaching lessons can also be a valuable reflective tool for all teachers and is especially revealing for preservice teachers. In addition to taping teaching sessions, another best practice for reflection in action for preservice teachers involves using small groups. Groups of three to four preservice teachers are created as teaching teams to experience all facets of classroom teaching. Team members are able to collectively recall more of the events and sequences that occurred. Some teams even record notes or use the taping strategy for later debriefing. As the semester progresses and a well established reflective teaching routine is rooted, the 28
teams diminish from four members to individual efforts as confidence and experience grows. The following are the prompts used as the basis for encouraging reflective teaching and frame informal assessment of learning interactions. While instruction is occurring, teachers try to keep these items in mind. Reflective Teaching Are the students on task? Do the students appear to understand the concept? If not, what are my alternatives/resources for actively adapting and modifying? • Are my instructional strategies appropriate for all students? • What are my expected outcomes? • What modifications are needed for reteaching? • •
Guided reflection. Post-teaching reflection allows the cycle of reflection to continue and helps teachers develop a greater repertoire of learning experiences. Too often, teachers are not encouraged to reflect due to time constraints. Time for reflection is imperative for developing teacher knowledge in novice teachers and in furthering the depth of knowledge for experienced teachers. By looking deeper into the learning interaction, going beyond the superficial, reflection provides teachers with insight into their teaching successes and failures. Reflection affords teachers the opportunity to select best practices for specific contexts and specific students. In methods course experiences, debriefing after teaching with preservice teachers is an extremely valuable strategy. Debriefing for inservice teachers may be achieved in collaborative efforts within graduate classes or with colleagues in schools. Developing an individual system of responding to the prompts discussed next is important. These prompts are similar to the ones initially addressed in the reflective planning section and have been used to begin classroom debriefing experiences for teachers, but notice the additional aspects addressed here. Teachers are asked to use specific examples from their teaching experiences to support, clarify, and elaborate responses. Guided Reflection Write a guided reflection. The Lesson • What were the goals of the lesson? • What did you do to make this lesson relevant? • What changes did you make in the flow of the lesson? Why and when? Reflection
What strategies (if any) were used for remediation?
The Learner • What questions were asked, how did you respond, and why were they asked? • If no questions were asked, reflect and respond to this. • What motivated student learning the most and why or why not? • What did you learn about students’ prior understanding and approaches to the content? The Teacher • How did you motivate and keep the learners on task? • What type of questioning did you use? • How did you assess the students’ learning and the success of the lesson? • What would you do differently? How and why? *Add other “teacher thinking”. Expert teachers may regard many of these reflective questions as normal components of classroom instruction. It is important, however, for all teachers to view these strategies in a new light. Learners are better served, as reflection becomes a habit of mind for teachers. Devoting valuable time to writing post-teaching reflections is a positive action towards becoming a reflective teacher. Reflective teaching has the potential to directly impact instruction and student achievement. This, in turn, reveals the power of reflective thinking and teaching. Level Three: Developing a Reflective Venue Journal writing is a popular instructional strategy for teachers to use in the classroom with students. Often teachers overlook its value as a reflective tool for teaching. To initiate and maintain a professional reflective journal in teaching is quite rewarding. Once preservice and inservice teachers develop through levels one and two, level three offers teachers an opportunity to develop a practical and individualized approach to reflective teaching. Reflective journal entries enable teachers to track thinking and learning, to evaluate these processes, and to improve teaching. Entries are more than recording and reporting events. They are visions of teaching and learning experiences recreated through thinking, feeling, and intuition. Establishing a journal writing routine is revealing and rewarding. Effective reflective journal writing provides evidence of and documents the details of how teachers plan, prepare, execute, and evaluate the teaching and Norene Vail Lowery
learning tasks. It encourages goal setting and documents the process as well as the products of the teaching experience. Teachers think about their own learning and understanding of the subject matter. Journals are the venue for implementing the reflective teaching strategies advocated in this discussion. Teachers are encouraged to begin journal entries at the beginning of the semester by addressing the prompts on reflection, on prior experiences in mathematics, and on mathematics and teaching from the first of this discussion. As reflective journal writing becomes a natural part of teaching, teachers may find a free response format more appropriate. This format is also encouraged as a basic venue, since journal writing can be quite overwhelming for novice teachers. Free Response Format • What did the class do today? • What did I expect students to learn? • What did I learn? Give some thought to what you and the students learned from the activities. A brief description of what was done in class that day helps to reflect on the learning. Was it relevant? Too easy? Too hard? What did you learn about mathematics, about teaching, about students, and about yourself? Levels one and two, understanding the importance of reflective thinking and implementing reflective strategies, are core components of both my undergraduate and graduate courses. Reflective practices are used with preservice teachers as they plan mathematics lessons to be taught in an elementary, field-based context during a semester-long, mathematics methods course. Inservice teachers practice these same exercises with their own mathematics classrooms, and then report insights in the context of their university-based graduate methods course. As time and classroom experience progresses, teachers find individual and unique ways to use journal writing as a reflective tool to improve instruction and to increase student understanding and achievement. Reflective Summary Although the role of reflection in teaching is considered important, reflective action in teachers is inhibited often by isolation of teachers or by the structure of courses and schools. The promotion of reflection should begin early in preservice experiences and with novice teachers (Artzt, 1999). Expert teachers are encouraged to share reflective practices with colleagues. Research findings confirm that the 29
likelihood of long-term success for many novice teachers is hindered by the absence of expert guidance, support, and opportunities to reflect (Veenman, 1984). Through reflective teaching, all teachers acquire critical skills in determining the value of instructional strategies, in assessing students’ mathematical understanding, and in developing curricular knowledge. School administrators, teacher educators, and expert teachers play pivotal roles in supporting the development of reflective teachers. Successful classroom experiences of preservice and inservice teachers demonstrate the value of reflective teaching. Reflective teaching practices promote greater student achievement and success in the classroom. Benefits from reflective teaching include increases in confidence, autonomy, and self-efficacy for teachers (Lowery, 2002). More effective questioning techniques – such as use of those that promote higher order thinking skills and use of openended questions – are employed, and classroom discourse is enhanced. Reflection allows teachers to judge mathematics grade-level appropriateness, to assess student abilities, to evaluate the use of motivational techniques, and to design appropriate and challenging mathematical learning activities. Reflective teaching is an essential skill for teachers and is a powerful component of successful teaching (Goodell, 2000; Mewborn, 2000). The power of reflection that blossoms from implementing reflective strategies strengthens the teaching and learning of mathematics. However, time is a critical factor. Time spent on developing reflective thinking and teaching is time well spent (Artzt & Armour-Thomas, 1999). The depth of understanding revealed in reflective teaching, the resulting improvement in instruction, and the ultimate growth in student learning far surpasses the initial sacrifice of time. Admittedly, this discussion asks for a lot from teachers. Yet, the best practices that have been presented here have merit. Teachers, preservice and inservice alike, have reported gaining much from reflective practices (Lowery, 2002). Reflective teaching in mathematics reaps incredible rewards. Inservice teachers report having taken the next step in promoting reflective practice by their mathematics students. Having students develop reflective perspectives on learning mathematics is promoted in the NCTM’s (2000) principles and standards. Students benefit by reflecting on their own learning to make sense of mathematics. Reflection is a crucial component that must be incorporated into every teacher’s toolkit of instructional strategies at all levels 30
of mathematics instruction and learning. To provide quality instruction and to increase student success and achievement in all classrooms, the basic three “R’s” must include a fourth—Reflection. REFERENCES Artzt, A. (1999). A structure to enable preservice teachers of mathematics to reflect on their teaching. Journal of Mathematics Teacher Education, 2(2), 143–166. Artzt, A., & Armour-Thomas, E. (1999). A cognitive model for examining teachers’ instructional practice in mathematics: A guide for facilitating teacher reflection. Educational Studies in Mathematics, 4(3), 211–235. Borko, H., & Livingston, C. (1989). Cognition and improvisation: Differences in mathematics instruction by expert and novice teachers. American Educational Research Journal, 26(4), 473–498. Borko, H., & Shavelson, R. (1990). Teachers’ decision-making. In B. Jones & L. Idols (Eds.), Dimensions of thinking and cognitive instruction (pp. 311–346). Hillsdale, NJ: Erlbaum. Brooks, J. G., & Brooks, M. G. (1993). In search of understanding: The case for constructivist classrooms. Alexandria, VA: Association for Supervision and Curriculum Development. [Preface, pp. vii–viii, by Catherine Twomey Fosnot]. Carter, K., Cushing, K., Sabers, D., Stein, P., & Berliner, D. (1988). Expert-novice differences in perceiving and processing visual classroom stimuli. Journal of Teacher Education, 39(3), 25–31. Cooney, T. (1985). A beginning teacher’s view of problem solving. Journal for Research in Mathematics Education, 16, 324–336. Dewey, J. (1933). How we think. Boston: Heath. Elbaz, F. (1983). Teacher thinking: A study of practical knowledge. New York: Nichols. Feiman-Nemser, S., & Buchmann, M. (1985). Pitfalls of experience in teacher preparation. Teachers’ College Record, 87(1), 53–65. Fenstermacher, G. D. (1994). The knower and the known: The nature of knowledge in research on teaching. In L. DarlingHammond (Vol. Ed.), Review of Research in Education (Vol. 20, pp. 3–56). Washington, DC: American Educational Research Association. Frid, S. (2000). Constructivism and reflective practice in practice: Challenges and dilemmas of a mathematics educator. Mathematics Teacher Education and Development, 2, 17–33. Glaser, B., & Strauss, A. (1967). The discovery of grounded theory. Chicago: Aldine. Goodell, J. (2000). Learning to teach mathematics for understanding: The role of reflection. Mathematics Teacher Education and Development, 2, 48–61. Hersch, R. (1986). Some proposals for reviving the philosophy of mathematics. In T. Tymoczko (Ed.), New directions in the philosophy of mathematics (pp. 9–28). Boston: Birkhäuser. Hoberman, S., & Mailick, S., (1994). (Eds.) Professional education in the United States: Experiential learning, issues, and prospects. Westport, CT: Praeger. Lortie, D. (1975). Schoolteacher: A sociological study. Chicago: University of Chicago Press.
Lowery, N. V. (2002). Construction of teacher knowledge in context: Preparing elementary teachers to teach mathematics and science. Journal of School Science and Mathematics Association (SSMA), 102(2), 16–31.
accommodating prior knowledge. The following selected passages may help to make evident what a constructivist learning approach means, for me as the teacher with the intent to create such a classroom:
Mewborn, D. (2000). Learning to teach elementary mathematics: Ecological elements of a field experience. Journal of Mathematics Teacher Education, 3(1), 27–46.
Constructivism is not a theory about teaching. It's a theory about knowledge and learning. Drawing on a synthesis of current work in cognitive psychology, philosophy, and anthropology, the theory defines knowledge as temporary, developmental, socially and culturally mediated, and thus, non-objective. Learning from this perspective is understood as a self-regulated process of resolving inner cognitive conflicts that often become apparent through concrete experience, collaborative discourse and reflection (Brooks & Brooks, 1993, p. vii).
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author. Nisbett, R. E., & Wilson, T. D. (1977). Telling more than we can know: Verbal reports on mental processes. Psychological Review, 84(3), 231–259. Schön, D. (1987). Educating the reflective practitioner: Toward a new design for teaching and learning in the professions. San Francisco, CA: Jossey-Bass. Tisher, R. (Ed.). (1978). The induction of beginning teachers in Australia. Melbourne: Monash University.
The constructivist viewpoint of learning supports reflective thought (Frid, 2000).
Tobin, K., Tippins, D., & Gallard, A. (1994). Research on instructional strategies for teaching science. In D. Gable (Ed.), Handbook of research on science teaching and learning: A project of the National Science Teachers Association (pp. 45–93). New York: Macmillan.
Prior knowledge influences knowledge construction. As a reflective tool, constructivism enables teachers to design appropriate learning activities, promotes higher levels of thinking about educational problems (reflection), and leads to questioning, and the construction of more knowledge (Tobin, Tippins, & Gallard, 1994).
Veenman, S. (1984). Perceived problems of beginning teachers. Review of Educational Research, 54(2). 143–178. Wilson, S. M., Shulman, L. S., & Richert, A. E. (1987). “150 different ways” of knowing: Representations of knowledge in teaching. In J. Calderhead (Ed.), Exploring teachers’ thinking (pp. 104–124). London: Cassell. 1
I choose the word constructivist here to represent perspectives that learning is constructed by a learner who is developing new knowledge, based upon as well as
Norene Vail Lowery
Here, I refer to the vision for teaching and learning set forth in the National Council of Teachers of Mathematics Principles and Standards for School Mathematics (2000).
The Mathematics Educator 2003, Vol. 13, No. 2, 32–37
Impact of a Girls Mathematics and Technology Program on Middle School Girls’ Attitudes Toward Mathematics Melissa A. DeHaven Lynda R. Wiest This research investigated the impact of an all-female, non-school-based mathematics program on middleschool-aged girls’ attitudes towards mathematics. Girls who attended a Girls Math and Technology Program for two consecutive years completed the Modified Fennema-Sherman Mathematics Attitude Scale before and after attending the program. Confidence scores increased significantly, whereas score increases in perceived usefulness of mathematics and perceived teachers’ attitudes toward the girls in mathematics were not significant. (The “mathematics as a male domain” subscale was not assessed due to a low reliability score.) Race and community background factors did not significantly affect the girls’ scores. Implications of findings and key program features are discussed.
During their early years, students develop the skills and attitudes toward learning that form the basis for future academic growth (Boland, 1995). If students develop a negative learning pattern toward a subject, it is extremely difficult to change. Females’ lower mathematics achievement in comparison with males is one area of educational concern that appears to be attitudinally based. On the 2000 National Assessment of Educational Progress (NAEP), males attained higher scores than females at the three grade levels tested (fourth, eighth, and twelfth). Males’ Scholastic Assessment Test–Mathematics (SAT-M) scores for the 2000-2001 school year topped that of females by 35 points (National Center for Education Statistics, 2002). Fox and Soller (2001) point out that performance differences on the SAT-M, which also appear on the Graduate Record Exam (GRE), can be costly for women in terms of college admissions and scholarship decisions. The research reported here investigated whether voluntary participation in a Girls Math and Technology Program improved middle-school-aged girls’ attitudes toward mathematics. Results are reported and discussed for a group of two-year program participants’ initial and follow-up ratings of their personal confidence in mathematics, perceived Melissa A. DeHaven teaches third grade at Smithridge Elementary School in the Washoe County School District in Reno, Nevada. She recently completed her master’s degree in elementary education with an emphasis in mathematics at the University of Nevada, Reno. Lynda R. Wiest is an Associate Professor of Mathematics Education at the University of Nevada, Reno. Her professional interests include K-8 mathematics education, educational equity, and teacher education. 32
usefulness of mathematics, and perceptions of their regular classroom teachers’ attitudes toward themselves in mathematics. The data were also examined for Whites and Non-Whites, as well as rural and urban participants, to see if the program had a differential impact on some participants. Review of Related Literature Attitudes toward mathematics, including perceptions of how appropriate mathematics is for females, play a prominent role in females’ lower performance and participation in mathematics in relation to males. Based on their analysis of NAEP data trends, Bae, Choy, Geddes, Sable, and Snyder (2000) contend, “Achievement gaps appear more closely related to attitudes than to course taking” (p. 117). The data show that females are less likely than males to like or to think they were good at mathematics. Females also experience mathematics anxiety to a greater degree than males (Levine, 1995). Females’ dispositions toward—and hence achievement and participation in—mathematics are believed to be socialized, inculcated by a society that tends to view mathematics as a male domain and which perpetuates the idea that males are naturally more mathematically inclined (Hanson, 1997). Teachers sometimes contribute to girls’ poor self-concept in mathematics. They may imply, for example, that girls do not need mathematics or they may react more negatively when girls ask questions of clarification than when boys ask (Jackson & Leffingwell, 1999). Jones and Smart (1995) consider lack of confidence to be a major factor affecting girls’ low participation in mathematics. Much interest in single-sex educational settings has appeared in recent years. Evidence from a variety of Impact of a Girls Mathematics Program
researchers and educators speaks to increased confidence, achievement, or subsequent participation in higher-level coursework for girls in single-sex mathematics classrooms (e.g., Streitmatter, 1997; Wood & Brown, 1997). Participants in Streitmatter’s (1997) two-year study of seventh- and eighth-grade girls in all-female mathematics classes reported an enhanced ability to learn the mathematics, an improved view of themselves as mathematicians, and a clear preference for this type of environment. One reason for girls’ greater comfort level in this type of classroom may be their expressed concerns about intimidation by boys in mixed-gender mathematics settings, namely, fear of being dubbed smart or fear of asking questions that boys deem “dumb” or otherwise unacceptable (Durost, 1996). Moreover, boys tend to dominate classroom conversation, be called on in class, be permitted to call out in class more often than girls, and they receive more teacher attention, including more useful feedback (Durost, 1996; Sadker, Sadker, Fox, & Salata, 1993/94). Numerous out-of-school Science, Mathematics, Engineering, and Technology (SMET) programs for girls, such as after-school clubs or summer programs, have had a positive impact on their participants in terms of knowledge acquired and—in particular—favorable attitudes gained (e.g., Karp & Niemi, 2000; Mawasha, Lam, Vesalo, Leitch, & Rice, 2001). Dobosenski (2001) maintains that these types of experiences should begin in elementary or early middle school. Common elements in successful SMET programs include: a comfortable learning climate (e.g., fun, noncompetitive, open to questions); career-related information and issues; development of SMET content knowledge acquired experientially; academic and social support that includes peers and adult role models; self-concept and confidence building through effective group work and successful performance in SMET activities (Campbell, 1995; Mawasha et al., 2001). Opportunities to see mathematics as femaleappropriate permeate these program features. Girls’ interest in mathematics begins to wane at about the middle school level, which is also the juncture at which students make decisions about future course enrollments and career tracks. Therefore, middle school is a critical “make-or-break-it” point for girls in mathematics (Campbell, Denes, & Morrison, 2000). Researchers stress the importance of offering early intervention programs for underrepresented groups (e.g., girls and students of color). These programs would emphasize career preparation, improve mathematics skills, and develop interest and Melissa A. DeHaven & Lynda R. Wiest
positive attitudes (Trentacosta & Kenney, 1997). The program this paper describes is one attempt to bolster and extend middle school girls’ in-school mathematics experiences. The Girls Math and Technology Program The Girls Math and Technology Program1 is available to Northern Nevada girls who will enter grade 7 or grade 8 the fall after they enter the program. The main program component is a five-day, residential summer camp held at the University of Nevada, Reno with classes held at the College of Education. The program includes two full-day Saturday sessions, one held in the fall and one in the spring of the following school year. The program began in the summer of 1998 and ran for four years by the time data analysis took place for this research. Currently, applications are sent to all Northern Nevada public, private, and Native American schools. The typical class size is 28 girls who work with others of their own grade level. In 1998, 28 girls entering grade 7 participated in the program, followed by 56 girls entering grades 7 or 8 in 1999, 76 girls entering grades 7, 8, or 9 in 2000, and 57 girls entering grades 7, 8, or 9 in 2001. The girls are randomly selected to ensure a fair selection process, so students of varied ability, race/ethnicity, socioeconomic status, and community background have the opportunity to participate. Scholarships are available to participating girls with demonstrated financial need. The mathematics topics addressed during the Girls Math and Technology Program include geometry, algebra, data analysis and probability, problem solving, and spatial skills. The girls also learn biographical information about historical and contemporary female mathematicians, and a guest speaker from the local community discusses her use of mathematics on the job. Two to four all-female staff members are in each classroom at all times. The staff consists of a balanced mix of veteran teachers who are active in mathematics education and upper-division teacher education majors or beginning teachers. Each lesson is developed in accordance with the Nevada Mathematics Standards established for the grade level the girls will enter in the fall. Two key program components designed to impact participants’ attitudes positively include providing female role models and employing an instructional approach that involves hands-on, conceptual, collaborative learning in a non-threatening atmosphere. Although these features can be incorporated into the 33
regular classroom, they appear to be infrequent or at least inconsistent aspects of middle-grades mathematics instruction, as the girls’ comments indicated in other research on this program (Wiest, 2003). Moreover, the single-sex nature of the program—in terms of both participants and staff—deviates from the typical mathematics classroom and was perceived to support the two program components noted above. Further discussion of critical program elements appears in the Discussion and Summary section of this paper. Research Purpose The purpose of this research was to investigate the impact of a same-sex, non-school-based mathematics program on middle-school-aged girls’ perceptions of their attitudes towards mathematics. 1. Did the girls’ perceptions of their personal confidence in mathematics, the usefulness of mathematics, mathematics as a male domain, and their teachers’ attitudes towards themselves in mathematics improve over time after attending the Girls Math and Technology Program for two consecutive years? 2. Were the girls’ attitudes influenced by their race (White or Non-White) or community background (urban or rural)? Research Method Sample The research sample consists of 36 Northern Nevada girls who attended the Girls Math and Technology Program for two consecutive years. Each girl had started the program during the summer prior to entering grade 7 and returned a year later prior to entering grade 8. The girls’ backgrounds are varied in terms of mathematics ability, socioeconomic status, and home community type. The sample includes 64% Whites and 36% Non-Whites (5% Black, 20% Native American, 5% Hispanic, 3% Asian, and 3% Biracial), of which 61% come from an urban area and 39% from a rural area. Design and Procedures The data-gathering instrument used in this research was the Modified Fennema-Sherman Mathematics Attitude Scale. This scale provides information about girls’ attitudes toward mathematics in the following categories: personal confidence about the subject matter, usefulness of the subject matter, perception of
the subject as a male domain, and perception of teachers’ attitudes toward the respondent in the subject2. The Modified Fennema-Sherman Mathematics Attitude Scale contains 47 positive and negative statements on a five-point, Likert-type scale that ranges from “strongly agree” to “strongly disagree.” The highest score possible is 235, with 5 points assigned to the most self- or mathematicsfavorable choice on each of the 47 items. The confidence, usefulness, and teacher’s attitudes subscales each contain 12 items with a highest possible score of 60. The male domain subscale contains 11 items with a highest score of 55. The Modified Fennema-Sherman Mathematics Attitude Scale was given to the girls the first day (pretest) of the Girls Math and Technology Program as well as the final day (post-test) of the week-long summer camp they attended for the first and second years, respectively. Data Analysis Pre-test and post-test scores on the FennemaSherman instrument were used for each girl in this sample. The scores consisted of totals for each of the four subscales. A reliability analysis, using Cronbach’s alpha, was conducted to test for internal consistency within each of the four subscales for the pre- and post-tests. Means and standard deviations were calculated for the four subscales for the pre- and post-test. To determine if the subscale scores improved, the means for the two tests in each of the four subscales were compared using a two-tailed, paired-samples t-test, with alpha set at the .10 level. To determine whether the girls’ attitudes were related to their race (White or Non-White) or community background (urban or rural), we conducted a one-way analysis of covariance (ANCOVA). Means and standard deviations were calculated for each level of the variables in all subscales for both pre- and posttests. Results Internal consistency for each subscale was calculated using Cronbach’s Alpha. For the pre-test, the highest alpha (.79) was obtained for teachers’ attitudes, with .75 for confidence, .74 for usefulness, and .37 for male domain. For the post-test, the highest alpha (.87) was obtained for confidence, with .84 for teachers’ attitudes, .81 for usefulness, and .61 for male domain.
Impact of a Girls Mathematics Program
Table 1 Modified Fennema-Sherman Mathematics Attitude Scale: Pre- and Post-Test Means and Standard Deviations Confidence
Discussion and Summary
The alphas for three of the subscales—confidence, usefulness, and teacher’s attitudes—fell within the acceptable range of .70 or above. However, the reliability of the male domain subscale was below the acceptable range, with the pre-test analysis at .37 and the post-test at .61. Therefore, it was omitted from further exploration.
The most influential aspect of the Girls Math and Technology Program is the positive impact it has on the girls’ self-confidence in mathematics (and perhaps technology, which the instrument did not measure). Girls’ and boys’ confidence in their mathematics abilities do not differ in the early grades, but a lack of confidence becomes evident for girls as they enter middle school (Boland, 1995). This is particularly important because mathematics becomes more complex at this point in time (Boland, 1995). Improvements in participants’ perceptions of the usefulness of mathematics and their teachers’ attitudes toward themselves were slight and were not statistically significant. In the case of mathematics’ utilitarian value, scores were already somewhat high and thus were less likely to show a statistically significant increase. It may also be difficult for middle school students in general to appreciate the usefulness of mathematics, because they may be too old to accept rhetoric stating that mathematics is useful and they may be too young to associate school mathematics with their daily lives in a meaningful manner. It is disappointing that the girls’ greater self-confidence in mathematics, as associated with this program, did not translate into more positive perceptions of their teachers’ attitudes toward themselves. This may highlight educators’ critically important role—and therefore the need for high-quality professional
Girls’ Attitudes Toward Mathematics A paired-samples t-test was conducted to evaluate if the girls’ attitudes improved over the two years they attended the camp. Table 1 shows mean and standard deviation scores for the two Fennema-Sherman Mathematics Attitude Scale tests: the pre-test at the beginning of the first year and the post-test at the end of the second year. The results show that the increase in the girls’ confidence level was statistically significant (t (35)=2.65, p= .012). The girls’ scores on the other two subscales did not increase significantly over the two-year period. Race and Community Background A one-way analysis of covariance (ANCOVA) was conducted on each subscale to evaluate if the girls’ scores were influenced by their race and by their community background with the pre-test scores as the covariate. Race and community background factors did not significantly affect the girls’ scores (see Table 2).
Table 2 Modified Fennema-Sherman Mathematics Attitude Scale: Analysis of Covariance for Influence of Race and Community Background Confidence
Race X Area
Melissa A. DeHaven & Lynda R. Wiest
development in gender-equitable teaching—in mathematics instruction for middle school girls. It is not possible to determine from these data the impact perceptions of teachers’ attitudes had on the girls’ attitudes. Nevertheless, it is reasonable to assume that teachers, with whom students spend a great deal of time and who judge the value of students’ work, influence students’ academic self-perceptions. According to these data, differences in race or community background did not appear to cause differential program impact. Instead, several shared attributes and interests predominated across the varied individuals who participated in this program. In other research on this program (Wiest, 2003), qualitative data in the form of personal interviews, camp-end surveys, and fall follow-up questionnaires showed some of these key commonalities to be gender, interest in mathematics, and a chance to meet new people and experience life away from home. Several features led to this program’s success (Wiest, 2003). The residential nature of the summer camp is a critically important aspect in that it allows girls from rural towns to participate. For instance, a group of parents drive their daughters seven hours from a remote Native American reservation to Reno in order to attend this program. By staying over night, the girls bond with each other and with the staff members over the course of the week, and they see that girls and women who like mathematics are “normal” people with many interests and abilities. One second-year camper’s parent told the camp director, for example, that one thing that surprised and impressed her daughter the previous year was that the camp director had played football with the girls on the campus quad during an evening recreation time. Another important program element, according to Wiest’s (2003) qualitative data, is the type, amount, and quality of the mathematics content. The topics addressed in the program are made to be interesting and challenging in ways many girls have never experienced. Several girls noted that learning mathematics without homework or the pressure of earning a grade greatly reduced their anxiety compared with their school experiences. This program’s instructional approach, which centers about group work and hands-on learning in a supportive environment, also surfaced as a key program element (Wiest, 2003). The program’s methods of instruction and comfortable climate seemed to differ from that which many of the girls encountered in school, and they better suited the girls’ needs and learning styles. 36
Finally, Wiest (2003) found that the all-female staff is another strong component of the Girls Math and Technology Program. The role-model aspect of this program, including female instructors, a female guest speaker, and information about accomplished female mathematicians, helped the girls see themselves as potentially successful mathematicians in both the present and future. Several parents said that their daughters began talking about the importance of mathematics and considering mathematics-related careers in the months after the summer camp had ended. This supplementary program for girls has the luxury of several benefits that the typical school does not. One is that instructors are chosen from among the most highly qualified local mathematics teachers. Another benefit not afforded to most schools—besides the single-sex nature of the participants and instructors—is the lower student-to-teacher ratio, with 2 to 4 instructors per 28 girls. This allows for more individual attention than most schools are able to provide. Closing Comments The National Council of Teachers of Mathematics (1995) notes that equity is a critical factor in the nation’s economic viability. The workplace requires that all Americans, including minorities and women, have the mathematics skills needed to meet the demands of the global marketplace. Eliminating the social injustices of past schooling practices will require the support of policymakers, administrators, teachers, parents, and others concerned about excellence and equity in mathematics education. All children can learn challenging mathematics with appropriate support and an equitable learning environment, regardless of ethnicity, race, gender, or social class. Until females and other lower-achieving and underrepresented students attain parity in mathematics, supplemental programs such as the one discussed in this paper can provide important support mechanisms beyond that which schools may offer. REFERENCES Bae, Y., Choy, S., Geddes, C., Sable, J., & Snyder, T. (2000). Trends in educational equity of girls and women. Education Statistics Quarterly, 2(2), 115–120. Boland, P. (Ed.). (1995). Gender-fair math. Newton, MA: WEEA Publishing Center. Campbell, G., Denes, R., & Morrison, C. (2000). Access denied: Race, ethnicity, and the scientific enterprise. New York: Oxford University Press.
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Campbell, P. B. (1995). Redefining the “girl problem in mathematics.” In W. G. Secada, E. Fennema, & L. B. Adajian (Eds.), New directions for equity in mathematics education (pp. 225–240). New York: Cambridge University Press. Dobosenski, L. (2001). Girls and computer technology: Building skills and improving attitudes through a girls’ computer club. Library Talk, 14(4), 12–16. Durost, R. A. (1996). Single sex math classes: What and for whom? One school’s experiences. NASSP Bulletin, 80, 27–31 Fox, L. H., & Soller, J. F. (2001). Psychosocial dimensions of gender differences in mathematics. In J. E. Jacobs, J. R. Becker, & G. F. Gilmer (Eds.), Changing the faces of mathematics: Perspectives on gender (pp. 9–24). Reston, VA: National Council of Teachers of Mathematics. Hanson, K. (1997). Gender, discourse, and technology. Newton, MA: Education Development Center. Jackson, C. & Leffingwell, R. (1999). The role of instructors in creating math anxiety in students from kindergarten through college. The Mathematics Teacher, 92(7), 583–586. Jones, L., & Smart, T. (1995). Confidence and mathematics: A gender issue? Gender and Education, 7, 157–166. Karp, K. S., & Niemi, R. C. (2000). The math club for girls and other problem solvers. Mathematics Teaching in the Middle School, 5, 426–432. Levine, G. (1995). Closing the gender gap: Focus on mathematics anxiety. Contemporary Education, 67, 42–45. Mawasha, P. R., Lam, P. C., Vesalo, J., Leitch, R., & Rice, S. (2001). Girls entering technology, science, math and research training (GET SMART): A model for preparing girls in science and engineering disciplines. Journal of Women and Minorities in Science and Engineering, 7, 49–57. National Center for Education Statistics. (2002). Digest of education statistics, 2001. Retrieved May 2, 2002, from http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2002130
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National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics. Reston, VA: Author. Sadker, M., Sadker, D., Fox, L., & Salata, M. (1993/94). Gender equity in the classroom: The unfinished agenda. The College Board Review, 170, 14–21. Streitmatter, J. (1997). An exploratory study of risk-taking and attitudes in a girls-only middle school math class. The Elementary School Journal, 98, 15– 26 Trentacosta, J., Kenney, M. (Eds.). (1997). Multicultural and gender equity in the mathematics classroom: The gift of diversity: 1997 yearbook. Reston, VA: National Council of Teachers of Mathematics. Wiest, L. R. (2003). Impact of a summer mathematics and technology program for middle school girls. Manuscript submitted for publication. [Contact firstname.lastname@example.org for the manuscript.] Wood, B. S., & Brown, L. A. (1997). Participation in an all-female Algebra 1 class: Effects on high school math and science selection. Journal of Women and Minorities in Science and Engineering, 3, 265–277 1
Interested individuals can obtain the manuscript entitled “Impact of a Summer Mathematics and Technology Program for Middle School Girls” by contacting Lynda Wiest at email@example.com. 2
Teachers’ attitudes refer to those of regular classroom teachers rather than instructors employed in this program. These attitudes center about teachers’ interest in, respect for, and encouragement of the girls as present and future mathematicians. For more information, see the FennemaSherman Mathematics Attitude Scale at http://www.woodrow.org/teachers/math/gender/ 08scale.html.
The Mathematics Educator 2003, Vol. 13, No. 2, 38–46
First-Time Teacher-Researchers Use Writing in Middle School Mathematics Instruction Drew K. Ishii This paper is a study of 4 middle school teacher-researchers who engage in action research projects for the first time, in which they incorporate writing activities as part of their instructional practices. Embedded in a professional development program with an emphasis on reform mathematics efforts, the teacher-researchers report to their research support group on their experiences with using writing. They used writing in order to improve classroom communication and state-mandated test scores. Recordings of conversations, written reflections, and other documents showed that they used various writing activities including journal writing, essays, problem solving, and the writing of stories. The teacher-researchers identify the major benefits of using writing to be the support of student thinking and the increase in student discourse. The teachers’ projects encouraged future ideas for instructional change. Part of this research was presented at the 2002 Annual Meeting of the Mid-Western Educational Research Association, Columbus, OH. October 17, 2002
This qualitative study investigates the experiences of four middle school mathematics teacher-researchers engaged in action research as part of a professional development program. The focus of this paper is to examine the experiences, practices, and issues that emerged from the teacher-researchers’ projects as they employed non-traditional writing activities in their mathematics classes. Action research is a practice by which teacherresearchers have the opportunity to learn from and about their teaching. Through this methodology teacher-researchers can reflect, evaluate, and learn not only about their teaching, but also from their students. Conducting action research projects allows teacherresearchers to reflect on their teaching and to explore issues of teaching and learning that are relevant to their lives. Engaging in action research can benefit all those involved in that it can bring self-renewal and increase efficacy, morale, and student performance (Sagor, 2000). Additionally, researchers reported that action research increases a sense of professionalism for the teacher-researcher (Elliot, 1991; Smith, Layng & Jones, 1996). The teacher-researchers around whose experiences this discussion revolves were involved in a professional development program at a major urban midwestern research university. This program served as a master’s degree program for some teachers and as Drew K. Ishii is a doctoral candidate in mathematics education at The Ohio State University in Columbus, Ohio. His research interests are writing in mathematics, mathematical discourse and communication, and representations. 38
a professional development program offering graduate credit for those either not pursuing a master’s degree, or those who had previously obtained a master’s degree. The premise of the program was for the teachers to implement innovative practices in their teaching that coincided with current educational reform and conduct an action research project with the support of doctoral students and faculty from the university. This culminated in a final paper centered on their research. Collaborative efforts between teacherresearchers and universities as well as professional development programs such as this one serve to aid in teachers’ pursuits of conducting research projects of their own, and thus create a life-long process of inquiry for the teachers (Raymond & Hamersley, 1995). Given what research says about using writing in mathematics, I set out to see what the use of writing looked like in the field from these teacher-researchers’ experiences. It was important to me that I get their perspectives on using writing in mathematics instruction. In keeping with a grounded theory research methodology, the data was approached without a priori research questions that would subsequently drive the data analysis. Instead, personal questions or inquiry issues provided the motivation to investigate the experiences of the teacher-researchers in this project: 1. What do the teacher-researchers hope to gain by using writing? 2. How do the teacher-researchers use writing in their teaching? 3. What benefits do the teacher-researchers see in using writing? Writing in Middle School
Why Writing? The use of writing assignments in school mathematics gained recognition with the “writing to learn” movement in the ‘80s and continues today as evidenced by the National Council of Teachers of Mathematics’ (NCTM) standards document, Principles and Standards for School Mathematics (2000). The call from NCTM to make communication an important facet in the mathematics classroom has led to an increase in instructional activities that encourage communication not only between teacher and student, but also among students. The Communication Standard (NCTM, 2000) includes being able to organize, communicate, analyze, and evaluate thoughts using the language of mathematics. An essential facet of communication is writing, which is used in just about every academic subject though rarely in mathematics classes. When used, communication through writing in mathematics classes generally takes two forms: (a) journal writing, or (b) expository writing assignments and activities. In journal writing, students reflect on some activity or respond to a prompt given by the teacher in order to solidify their thinking on some topic or concept. In expository writing, students use writing as an active part of the learning process with in-class writing activities or prompts aimed towards explanatory or expressive purposes. For instance, a writing prompt may require students to solve a mathematical problem and then explain their thinking or problem-solving processes. Expository writing activities allow students to use another avenue or representation in their mathematical learning, along with a period of reflection when deciding what to write. Expository writing assignments can be thought of similarly to what some (Birken, 1989; Powell & Lopez, 1989; Rose, 1989) call transactional writing, in which the assignment is meant to be read by someone other than the writer, usually a teacher. It is important to consider both uses of writing in mathematics because each activity has its own benefits depending on what the teacher wants to accomplish (Birken, 1989; Borasi & Rose, 1989; Cai, Jakabcsin & Lane, 1996; Drake & Amspaugh, 1994). The writing that students do in mathematics classes is quite different from other classes or disciplines since mathematics is presented as a heavily symbolic discipline. The bulk of student work in mathematics classes consists of symbol manipulation. The symbols are the language of mathematics and ordinary language is used to explain the mathematics. The use of writing to learn mathematics, however, tries to use writing in different capacities of the learning process. Keith Drew K. Ishii
(1989) offers several types of writing assignments such as: assessment of material, anticipation of new material, discussion, peer collaboration, revision, and evaluation. Birken (1989) suggests that writing can be used for informal in-class writing, homework problems that interpret or analyze, essay questions, and formal technical writing. Multiple-entry logs, another type of writing technique, combine journal writing with expository writing; students are asked to respond to a prompt or problem, then revisit their writing and thinking periodically to see how it changes over time, if at all (Powell, 1997). In trying to articulate their thoughts into words, students engaged in these types of assignments reflect and internalize. This process promotes further learning. Two studies (Pearce & Davison, 1988; Shield & Galbraith, 1998) approached student writing in a discourse analytic manner where they classified students’ writing in order to determine the elements present in student writing. They offer a more in-depth look into student writing in mathematics classes. In these studies, researchers recognized that the type of writing that occurs in mathematics classes is different than that of other disciplines, and thus needed to be examined further in order to assess the elements of student mathematical writing. In effect, they examined the writing that resulted from various types of writing assignments and discovered how students communicated their knowledge to the teacher. Shield and Galbraith (1998) analyzed 8th-grade students’ writing and developed a coding scheme for content of the writing in order to generate a model of student writing. In addition to developing the coding scheme, they compared the writing samples with the type of writing that occurred in the students’ textbook. They identified six features of the students’ writing: exemplar, goal statement, kernel, justification, link to prior knowledge, and practice exercises. The most common of these was exemplar in which students gave written descriptions of specific examples, diagrams, conventions, and graphs (p. 39). In comparison with the textbook, they found that studentwriting samples heavily reflected the same type of writing style: (a) a focus on procedures and algorithms with little elaboration, and (b) an authoritative tone (p. 45). Previously, in 1988 Pearce and Davison determined the amount, kinds, and uses of writing that teachers employ in junior high school mathematics classes. By looking at student samples and teacher interviews, they classified five types of writing activities: direct use of language (copying and transcribing information), linguistic translation 39
(translation of mathematical symbols into words), summarizing/interpreting (summarizing, paraphrasing, and making personal notations about material from texts or other sources), applied use of language (situations where a mathematical idea is applied to a problem context), and creative use of language (using written language to explore and convey mathematically related language) (p. 10). They found that the direct use of language activities were most frequently used. Research on writing in mathematics offers not only various methods of incorporation into instruction, but also the benefits from using such techniques. Borasi and Rose (1989) found that journal writing had a therapeutic effect on students, as well as increased learning of the material, and improved problem-solving skills. They also found that teachers benefit from using writing in that they are better able to provide feedback and make better evaluations of student learning or misconceptions. From this, there is potential for teachers to make long-term improvements in their instruction. Miller (1992) reported similar benefits for teachers utilizing impromptu writing prompts. By reacting to student writing, instructional practices were influenced when the teachers would re-teach, delay exams, schedule review sessions, and initiate discussion over misconceptions. This account of some of the research in communication and mathematics shows how writing can be beneficial for both the teacher and the mathematical learning of the students. In many of these studies, researchers partnered with schools in an effort to study the issues concurrent with the mathematics reform efforts. In a similar manner, based upon the examination of the current NCTM standards, the teacher-researchers with whom I worked in the professional development program sought to incorporate similar research ideas into their instruction and thusly into their action research projects. Methodology The Project and the Teachers I assisted in a professional development program1 to support 4 of the 13 mathematics teacher-researchers enrolled. These four were in the data collection stages of their research when I joined the effort. I provided regular guidance in their data collection and analysis efforts for their action research projects. I had also served as a support person for one of the teachers (Iris2) in the previous year. As a doctoral student in mathematics education, I was asked to participate in this program as part of the support team because my research interests (communication and mathematics) 40
and experience would be useful to some of the teacherresearchers involved in the project. As part of their participation in the program, all of the teachers in the program were assigned a support person, who was either a graduate student or a university professor. The support person helped with planning and implementing instruction and provided support and expertise in their action research endeavors. By the time I joined the support team, the teacher-researchers had been in the professional development program for just over one year with one year left to go. Teachers joined the program in order to learn more from and about their own teaching, and (for some) to work towards a masterâ€™s degree in education. Each teacher-researcher chose a topic and designed research questions they would investigate throughout the duration of the two-year program. The desire to change their teaching practices drove their research questions, which in turn provided a theme for their instruction for the two years of the program. For their action research projects, the four teachers discussed in this paper chose to implement writing in their mathematics classes using either journals or expository writing exercises such as those mentioned earlier. Three of the teachers taught sixth-grade mathematics while one taught eighth-grade mathematics. The sixth-grade teachers, Iris, Jean, and Amber, taught in urban schools, while the eighth-grade teacher, Joanne, taught in a suburban school. Amber was the only teacher of the four who was working towards her masterâ€™s degree. The other three were in the program to obtain graduate professional development credit. These four teachers individually have fewer than 10 years teaching experience. From their research proposals and from numerous conversations, three reasons resonated between the teacher-researchers indicating why they chose writing in mathematics as a focus for their research. First, writing is encouraged in mathematics education reform efforts. To the teacher-researchers, writing in mathematics was a practice they saw as novel and outside the realm of the traditional mathematics classroom. Second, writing is incorporated in the openresponse elements of the state proficiency exams. As with most school districts across the country, student performance on state exams is important, and these teacher-researchers saw the utilization of writing as a technique that would help prepare their students for the tests. Finally, the open-response questions on state exams were traditionally an area of the exams in which middle school students in their districts scored very
Writing in Middle School
poorly. Thus, the teacher researchers sought to improve students’ scores by focusing on writing tasks. Research Design In this study, I used qualitative methods to examine the experiences of the four teacherresearchers. Three types of data were collected: audio recordings of meetings and conversations, documents collected from the teacher-researchers, and my personal field notes. Each data analysis meeting for the teacher-researchers’ projects was audio recorded and field notes were taken during those meetings. Other conversations regarding the projects were audio recorded as well. The documents that were analyzed included their research proposals, reflections throughout the past year, open-ended surveys, and final papers. I analyzed these tapes and documents using principles of grounded theory (Charmaz, 2000). The emergent patterns and themes in the taped conversations were investigated further and triangulated with the documents (Janesick, 2000). In qualitative research methods, these types of documents are important data sources because they catalog the participants’ beliefs, values, and experiences (Marshall & Rossman, 1995), as they did throughout the two years of this program. Although I was a support person for Iris and had personally assisted with her action research project, for this study I limited the scope of the audiotaped data collected from our interactions to those that included the other three participants in order to be fair to all of the teacher-researchers. Since I started supporting all of the participants approximately 10 weeks before the end of their school year, and subsequently the end of their data gathering and analyzing, the conversations that were audiotaped occurred within the near-weekly meetings of those 10 weeks. The documents, on the other hand, were collected throughout their program by the director and given to me once I joined the support team. Once the project concluded and all data for this study was collected, I inductively analyzed all of the documents, including my field notes, for emerging patterns and issues. These fell within two general categories, research issues, and issues related to the use of writing. For the purposes of this discussion the research issues have not been included in the findings. I then listened to the tapes of our meetings and conversations with the intent of finding more evidence to support the long list of codes that were made from the patterns and issues obtained from the documents. After several iterations of this process, the codes that Drew K. Ishii
boasted the most support were further examined and developed into the theory that will be discussed. It is important to note that in keeping consistency with the principles of grounded theory, disconfirming data or negative cases were sought after, but were not found. Without discussing each teacher-researcher’s individual project, the proceeding discussion is limited to their experiences with implementing the writing in their classes, including their future research directions. Findings What They Hoped to Gain Each teacher-researcher began their academic year by writing a research proposal outlining their research plans for the year. These proposals were complete with research questions, methods, and proposed data collection and analysis. As mentioned before, an important reason for the four teacher-researchers to implement writing into their mathematics instruction was to improve the open-ended response questions on their students’ state proficiency tests. Joanne said, “I hope to change the way students feel about math, help students do better in math, and increase their mathematical understanding.” Amber echoed this sentiment by explaining that she wanted to supply her students with appropriate tools for approaching the extended response questions on the state exams. From past teaching experiences, she noticed that her students struggled on open response questions and sought to improve their scores. Similarly, Jean hoped her project would result in a change in students’ attitudes and improve the open-ended response question scores. For all of the teacher-researchers, seeing their students succeed in mathematics was important. But beyond that, seeing that problem solving is an important aspect of daily life both inside and outside the classroom, Iris and Joanne wanted to furnish their students with the necessary tools to help them in the future. Iris said, “I am looking for some way of making problem solving less threatening in general, [and] to help increase students problem solving capabilities. Joanne agreed saying, “I am hoping that through writing, communicating, students’ attitudes and conceptual understanding will improve.” All of the teacher-researchers not only wanted their students to do better on their tests, but also wanted to help their students learn the mathematics and make it less difficult. This concern for their students provided motivation for their projects. From their research proposals, in addition to the current literature on writing in mathematics (e.g., Borasi & Rose, 1989; Johanning, 2000; Jurdak & Zein, 1998), the teacher41
researchers’ concerns and goals were both appropriate and reasonable tasks. Their Writing Activities Joanne. The teacher-researchers implemented writing in a variety of ways ranging from journal writing to problem solving. These types of activities were similar to those activities found in the literature. Joanne used writing activities to start class, frequently using them as a warm-up exercise to focus the students’ attention on the mathematics of the day. She used writing prompts that were problem-solving in nature and insisted that students work individually ensuring that everyone attempted the problem. She often had students form small groups giving the opportunity for sharing their strategies and solutions with each other. This led to increased student participation and motivation. Since students spent time working on the problems, they were interested in sharing their work and seeing the various ways other students approached the problems. Even if students did not understand how to process the problem or get the answer, they could share how they set up their information and attempted to solve it. To facilitate students' writing, Joanne developed a problem-solving format called ODEAR (see Zupancic & Ishii, 2002), an acronym that helped the students organize their thoughts when writing. ODEAR consists of five elements: Organize, Define, Explore, Answer, and Reflect. When given a problem to solve, the students used the acronym to start and thoroughly answer their problems. Iris. Iris’s employment of writing in her classes was done primarily as in-class activities. She used a prescribed writing process similar to Joanne’s ODEAR. Iris’s problem-solving format, called EPSE (Explore, Plan, Solve, Evaluate), was a process prescribed by her district’s curriculum materials. She used a teacher’s supplement as a source for many of the problems she assigned. Typically, Iris gave a word problem on the board and had students solve and write individually. Occasionally they would compare their work with each other, but generally they worked alone. In one of her lessons, Iris gave a problem and let students work together in groups of four to five. The groups then presented their work to the entire class allowing everyone to see the different solutions. She reported that students really liked that lesson and she found it beneficial too because she immediately saw what they knew about the material. Generally, Iris gave her classes a few EPSE problems every week. The students kept these problems along with their class 42
notes in binders that she referred to as their portfolios. She eventually used some of the students’ portfolios as data for her action research project. Based upon our research meetings and from Iris’ reflective writings, she felt she had difficulties keeping up with evaluating her students’ writing. She said, “I wasn’t able to respond to their problems as well as I should have. I should have given them more feedback and let them give each other more feedback.” Time was something with which all of the teacherresearchers struggled, but Iris felt that it was the major struggle for her. Since Iris rarely allowed group sharing of writing in the same way that her colleagues did, her students received limited benefits from reflecting on their writing after receiving feedback, whether it be from her or from fellow students. Amber. Amber was the only teacher-researcher who used journal-writing activities. She used a journal format where she asked students to write about their feelings or attitudes, mathematical processes, and mathematical concepts. The students kept journals or notebooks as records of all of their writing. The students regularly shared their writings that focused on the mathematical procedures and content. Amber periodically collected and provided feedback addressing all the types of journal entries—affective, procedural, and conceptual. She reported that students would have benefited more from the journal-writing assignments had she been able to collect them and provide feedback more often. She said that keeping up on journals was difficult especially since the process was new to her; it was difficult to adjust to the time constraints and reorganize time usage. Even so, Amber did use the journal-writing assignments to have the students share with each other and provide peer evaluations. Jean. Jean used a variety of writing activities, instead of focusing on one type of activity as her colleagues did. She used writing activities both during class and as final thoughts or assignments that encouraged reflection and summarization. One activity in particular was what Jean referred to as the exit ticket, a final activity of the class period that required reflection or solving a problem. This activity was to be completed either before students left the class or moved onto science, which she also taught. She also used prewriting assignments for expository essays to help create assignments her students could share and edit together. This was a way to foster thinking ahead of time. Jean felt this along with students’ writing, sharing, and revising, could lead to clear cohesive pieces of expository writing. In addition to the well Writing in Middle School
thought-out prewriting and writing assignments, Jean used writing as a way of closing down or reflecting upon discovery-type activities. For instance, when she used manipulatives to model fraction arithmetic, she included a writing activity for students to express what they discovered. Jean also had a year-long project where her sixth-grade students made math story books for elementary school students from the neighboring elementary school. At the end of the year, Jean’s students shared their books with their partner class, and she brought samples in for the rest of the members of the professional development program to see. Their Observed Benefits In our final conversations, as well as in their reflective writings, the teacher-researchers’ concluded that after using the writing activities for a whole school year, there were two aspects of the experience that were of noteworthy benefit to the students and their learning. The greatest benefit was that the use of writing assignments promoted student-to-student discourse, something that usually does not occur in the traditional mathematics classrooms. The second benefit the teacher-researchers identified was an observed increase in student motivation, thinking, and understanding from previous years of teaching. This increase was a “perception” (sense of increase), not an empirical increase since teachers did not perform actual comparisons from the previous years. The teacher-researchers acknowledged that the benefits to students also served as benefits for themselves in that they saw overall improvement in the very things they sought to change. Discussion Improved discourse In reform mathematics efforts (NCTM, 2000), student discourse is an important element in the activities of the mathematics classroom. Current research supports the notion that social interactions whether they be whole-class or small group discussions benefit student learning (e.g., Cobb, Wood, & Yackel, 1993; Yackel & Cobb, 1996). Although improving student-to-student discourse was not a specific goal for the four teacher-researchers, they were well aware of the importance for increasing communication in general, and had that in mind when they chose to implement writing. In addition, increasing classroom communication was an overarching theme for the entire professional development program. The improvement in student discourse was somewhat of a surprise to the teacher-researchers in that it was not Drew K. Ishii
planned. For the students, however, it seemed as though discussions naturally followed their writing. Amber admitted that she never intended for the writing activities to accompany discussion of it among students. She planned to use writing as a learning tool students could use individually, and use the journals for personal reflection and learning. However, the discussion of her students’ writing began by accident when a student volunteered to read her writing aloud. Amber indulged the student and after a couple of instances, the student sharing of writing became a norm and expectation of the classroom activity. In a conversation we had about using writing and how student-to-student discourse seemed to be a natural consequence, Amber offered that the teacher would have to allow it. “I don’t necessarily think that employing a writing component in your math class is very beneficial unless you utilize it and discuss [the writing].” Amber also mentioned that in interviews with her students, they indicated it was not necessarily the writing that helped, but the sharing of the writing and the discussions that came after. Even the students saw the benefit of writing along with the opportunity to discuss what they wrote with each other. Iris commented that she agreed with what Amber discovered about writing in her classes. Iris’ goal was to improve students’ problem-solving skills, and she felt that writing alone would not be sufficient, but could when coupled with discussions of their problems and solutions. Though Iris did not use writing activities to promote discussions per se, she became aware that through discussion the writing might be used as a technique to encourage classroom discourse. Both Joanne and Jean reported that students enjoyed explaining their solutions to their classes and were often eager to share their findings with others. As a result, student participation became natural for students instead of requiring solicitation by the teacher. Jean responded, “The student-to-student discourse in my class has promoted conversation and debate about mathematical concepts.” Having students discuss and debate mathematical concepts is precisely the point of encouraging student discourse. Through those discussions students are given the opportunity to further reflect upon their own thinking while possibly augmenting other students’ thinking to their own. With regard to writing activities, students feel they have invested their time and effort into something other than ordinary mathematics work, and thus feel the natural progression to discussing their work with each other and their teachers. These conversations then provide the students with valuable 43
feedback about the way they are thinking about the mathematics. The writing activities do not have to end there however, another round of revision to the writing students have already produced can solidify thinking and add another layer to their understanding much the same way multiple-entry logs enable students to revisit their work (Powell, 1997). Supported student thinking Another consequence of using writing activities along with discourse is that it supports student thinking. Because of the reflective nature of speech and dialogue, discussions among students can be valuable tools for learning (Vygotsky, 1978). As mentioned earlier, the discussions that accompany writing activities enhance classroom communication and have the potential to provide students with another opportunity for reflection upon their thinking. Since writing is a product-oriented classroom activity, the students have a concrete record of their participation and of their thinking, which they can refer to and revise during discussions. The written product affords students the opportunity for critical reflection, which has the potential to give students control of their learning as well as a means of monitoring progress (Powell & Lopez, 1989). All of these steps within the activity of writing support students’ thinking in a way that is not usually seen in the traditional mathematics classroom. Thus the use of writing can provide students with extra tools for learning mathematics. Joanne agreed with this position saying that without discussion to “force” students to think about their thinking, the writing activities are not meaningful. She commented, “My students have learned many things from each other this year, and from themselves. Sometimes they understand better when another student explains the mathematics.” Joanne felt that if students really understand a concept they should be able to teach and explain it. Iris followed with a comment about argument and how it advances learning; “Trying to convince someone you are right through discourse is certainly a form of teaching and teaching is a great way to learn.” The relationship between learning and social interaction can be seen in Joanne and Iris’ experiences. The cycle of doing, thinking, and reflecting that writing promotes supports the learning process by empowering students so that they feel comfortable to take on peer teaching responsibilities. In Jean’s class, she noticed that reflection upon mathematical material did not necessarily have to take place in an elaborate/formal assignment, but could 44
occur as the day’s final activity. Recall that her exit ticket activity required students to work out a problem and/or reflect on it or that day’s lesson as a concluding activity for the day. Jean said, “The exit ticket at the end of the class lesson has encouraged students to think about what has been learned in class and encouraged discussion that sometimes does not occur in the classroom due to time.” Jean discovered that the students’ writing gave them a topic with which they could think deeply. Their ideas and thinking were pondered even after the class was over, and could provide an opening discussion for the next time they met. Writing also provided support for student thinking indirectly by supplying their teachers with feedback they would not normally have from their students. In a sense, student thinking was made more clear to their teachers, which in turn allowed the teacher-researchers to make adjustments in their teaching and acknowledge misconceptions. To this effect, Jean explained, “I sometimes realize that I may have not taught a concept clearly when many of the students have come to the same misdirected conclusion.” Amber concurred saying that she felt that she knew her students’ mathematical ability much better than in past years. “I know more about my kids than I ever have any other year,” demonstrating the ability of writing activities to transform learning experiences for students. Joanne remarked that she was able to find out what her students really knew, and cited an example of discovering that a poorly achieving student - knows more mathematics than his/her grades indicate. Future directions The benefits of using writing in their classes show that the teacher-researchers learned a great deal from their students by reading and participating in discussion. They learned from themselves by using different teaching techniques and deciding on better ways to foster student learning. They also learned a great deal from each other by participating in our conversations and meetings about the data analysis and debriefing of their action research projects. Another, among the many things the teacher-researchers learned not only about their students but also about themselves, is what they want and/or need to do in the next school year when they use writing. It is important to realize that when trying out new teaching techniques, everything might not result ideally the first time. Good teaching techniques take years to perfect, and these teacher-researchers have a sense of how they would proceed in the future. Writing in Middle School
Amber expressed that she wants to collaborate with her language arts teacher to use writing more than what she did this year. She also wants students to keep a journal book in the room instead of using loose paper as they did this year. Timely responses were a concern for Amber and she intends to make a better effort at responding in an appropriate amount of time. Joanne wants to try writing activities with her learning disability (LD) students. Seeing the benefit to her past year’s students, using writing with her LD students might show similar promise. She wants to have students grade their own and each other’s writing using the ODEAR rubric that she devised. Hearing about improved student discourse from the other teacherresearchers, Iris plans to incorporate the use of discourse with writing into her classroom. Next time she wants to incorporate more discourse and re-writing (post-writing) after they have discussed their solutions. Jean wants to make changes to the rubric she used to grade expository essays. This past year, she used the district’s rubric and ended up not liking it towards the end of the project. Concluding Thoughts After completing these projects with the teacherresearchers, I think they learned wonderful lessons from their own teaching. They enjoyed the process enough to want to continue the use of writing in their classes, and continue to make improvements in their teaching—one of the main goals of conducting action research in the first place. This research surveyed the experiences and issues that arose from first-time teacher-researchers incorporating writing strategies into their mathematics classrooms. Teacher-researchers utilized several types of writing strategies including expository writing, warm-up writing, problem solving, journal writing, and reflective writings. They discovered several benefits of using writing in their practice. They found that writing was not only advantageous to the students, but also to the teachers themselves. These results are consistent with research that addresses not only student benefits, but also those for teachers (Borasi & Rose, 1989; Miller, 1992). Students benefited from writing by increasing their thinking and reflection, and having an opportunity to share their writing that, in turn, led to dialogue and discussion with each other as well as the teachers. The teacher-researchers developed a better understanding of their students’ knowledge and conceptions because of the additional opportunities to discuss students’ thinking and provide feedback on their writing samples. The ultimate benefit from writing is that it Drew K. Ishii
enables more dialogue between all members of the classroom, something that is often missing from the traditional mathematics classrooms. This project served as a great learning tool for everyone involved. The teacher-researchers learned about their teaching, as well as potential future directions for their research. Action research provided another learning arena for teachers because they stepped back from their practice and evaluated it systematically. Furthermore, writing can serve as a learning tool that has the potential to be extremely beneficial as well as enjoyable when discussions are an integral part of the process. My involvement in this project gave me the opportunity to evaluate the use of writing in mathematics in action. Engaging in this project allowed me to see the applications of research to classroom situations and vice versa. Working with the four teacher-researchers highlighted the reality of conducting action-research in a middle school setting and all of the challenges and enjoyment that can result from it. This experience illustrated for me, firsthand, the issues and concerns teacher-researchers encounter when trying new teaching and instruction techniques for the first time. REFERENCES Birken, M. (1989). Using writing to assist learning in college mathematics classes. In P. Connolly, & T. Vilardi (Eds.), Writing to learn mathematics and science (pp. 33–47). New York: Teachers College Press. Borasi, R., & Rose, B. (1989). Journal writing and mathematics instruction. Educational Studies in Mathematics, 20(4). 327–365. Cai, J., Jakabcsin, M. S., & Lane, S. (1996). Assessing students’ mathematical communication. School Science and Mathematics, 96(5), 238–246. Charmaz, K. (2000). Grounded theory: Objectivist and constructivist methods. In N. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative research (2nd ed., pp. 509-535). Thousand Oaks: Sage. Cobb, P., Wood, T., & Yackel, E. (1993). Discourse, mathematical thinking, and classroom practice. In N. Minick, E. Forman, & A. Stone (Eds.), Education and mind: Institutional, social, and developmental processes (pp. 91–119). Oxford: University Press. Drake, B. M., & Amspaugh, L. B. (1994). What writing reveals in mathematics. Focus on Learning Problems in Mathematics, 16(3), 43–50. Elliot, J. (1991). Action research for educational change. Philadelphia, PA: Open University Press. Janesick, V. J. (2000). The choreography of qualitative research design: Minuets, improvisations, and crystallization. In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative research (2nd Ed., pp. 379–400). Thousand Oaks, CA: Sage.
Johanning, D. J. (2000). An analysis of writing and postwriting group collaboration in middle school pre-algebra. School Science and Mathematics. 100(3), 151–157.
Rose, B. (1989). Writing and mathematics: Theory and practices. In P. Connolly, & T. Vilardi (Eds.), Writing to learn mathematics and science (pp. 15–32). New York: Teachers College Press.
Jurdak, M., & Zein, R. A. (1998). The effect of journal writing on achievement in and attitudes toward mathematics. School Science and Mathematics, 98(8), 413–419.
Sagor, R. (2000). Guiding school improvement with action research. Alexandria, VA: Association for Supervision and Curriculum Development.
Keith, S. (1989). Exploring mathematics in writing. In P. Connolly, & T. Vilardi (Eds.), Writing to learn mathematics and science (pp. 134–146). New York: Teachers College Press. Marshall, C., & Rossman, G. (1995). Designing qualitative research. Thousand Oaks, CA : Sage.
Smith, S., Layng, J., & Jones, M. (1996). The impact of qualitative observational methodology on the authentic assessment process. Proceedings of Selected Research and Development Presentations (pp. 745–842). Indianapolis, IN: Association for Educational Communications and Technology.
Miller, L. D. (1992). Teacher benefits from using impromptu writing prompts in algebra classes. Journal for Research in Mathematics Education, 23(4), 329–340.
Shield, M., & Galbraith, P. (1998). The analysis of student expository writing in mathematics. Educational Studies in Mathematics, 36(1). 29–52.
National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics, Reston, VA: Author.
Vygotsky, L. (1978). Mind in society: The development of higher psychological processes. Cambridge: Harvard University Press.
Pearce, D. L., & Davison, D. M. (1988). Teacher use of writing in the junior high mathematics classroom. School Science and Mathematics, 88(1), 6–15. Powell, A. B. (1997). Capturing, examining, and responding to mathematical thinking through writing. The Clearing House: A Journal of Educational Research, Controversy, and Practices, 71(1), 21–25. Powell, A. B., & Lopez, J. A. (1989). Writing as a vehicle to learning mathematics: A case study. In P. Connolly, & T. Vilardi (Eds.), Writing to learn mathematics and science (pp. 157–177). New York: Teachers College Press. Raymond, A. M. and Hamersley, B. (1995, April). Collaborative action research in a seventh-grade mathematics classroom. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco, CA.
Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477. Zupancic, J. & Ishii, D. K. (2002) Writing as a tool for learning in mathematics: A case study in eighth-grade algebra. Ohio Journal of School Mathematics, 46(Autumn), 35–40. 1 The Teacher-Researcher Program was supported by grants under the federally funded Dwight D. Eisenhower Professional Development Program, administered by the Ohio Board of Regents, and The Ohio State University/Urban Schools Initiative funded through the Jennings Foundation. 2 All names are pseudonyms.
Writing in Middle School
The Mathematics Educator 2003, Vol. 13, No. 2, 47–57
Teachers’ Mathematical Beliefs: A Review Boris Handal This paper examines the nature and role of teachers’ mathematical beliefs in instruction. It is argued that teachers’ mathematical beliefs can be categorised in multiple dimensions. These beliefs are said to originate from previous traditional learning experiences mainly during schooling. Once acquired, teachers’ beliefs are eventually reproduced in classroom instruction. It is also argued that, due to their conservative nature, educational environments foster and reinforce the development of traditional instructional beliefs. Although there is evidence that teachers’ beliefs influence their instructional behaviour, the nature of the relationship is complex and mediated by external factors.
For the purpose of this paper, t e a c h e r s ’ mathematical beliefs refers to those belief systems held by teachers on the teaching and learning of mathematics. Educationalists have attempted to systematize a framework for teachers’ mathematical belief systems into smaller sub–systems. Most authors agree with a system mainly consisting of beliefs about (a) what mathematics is, (b) how mathematics teaching and learning actually occurs, and (c) how mathematics teaching and learning should occur ideally (Ernest, 1989a, 1989b; Thompson, 1991). Certainly, the range of teachers’ mathematical beliefs is vast since such a list would include all teachers’ thoughts on personal efficacy, computers, calculators, assessment, group work, perceptions of school culture, particular instructional strategies, textbooks, students’ characteristics, and attributional theory, among others. In this paper, the concept of progressive instruction is associated with a socio-constructivist view of teaching and learning mathematics. Socioconstructivism, which for the sake of brevity will be called just constructivism, gives recognition and value to new instructional strategies in which students are able to learn mathematics by personally and socially constructing mathematical knowledge. Constructivist strategies advocate instruction that emphasises problem-solving and generative learning, as well as reflective processes and exploratory learning. These strategies also recommend group learning, plenty of discussion, informal and lateral thinking, and situated learning (Handal, 2002; Murphy, 1997). In turn, Boris Handal has taught and lectured in schools and universities in Australia, Latin America and Asia. He has written extensively on academic issues in academic journals in the United States, United Kingdom, Australia, Latin America, and South East Asia. Boris obtained his Bachelors of Education from the Higher Pedagogical Institute of Peru, a Masters of Education from Edith Cowan University and his Doctorate of Education from the University of Sydney. In addition he has a postgraduate degree in educational technology from Melbourne University. Boris Handal
traditional instruction is associated with a behaviourist perspective on education. Behaviourist practices are said to emphasise transmission of knowledge and stress the pedagogical value of formulas, procedures and drill, and products rather than processes. Behaviourism also puts great value on isolated and independent learning, as well as conformity to established one-way methods and a predilection for pure and abstract mathematics (McGinnis, Shama, Graeber, & Watanabe, 1997; Wood, Cobb, & Yackel, 1991). Leder (1994) stated that in the behaviourist movement “the mind was regarded as a muscle that needed to be exercised for it to grow stronger” (p. 35). The study of teachers’ instructional beliefs and their influence on instructional practice gained momentum in the last decade. Some research on teachers’ thinking reveals that teachers hold wellarticulated educational beliefs that in turn shape instructional practice (Buzeika, 1996; Frykholm, 1995; McClain, 2002; Stipek, Givvin, Salmon, & MacGyvers, 2001; Thompson, 1992). Examples of research, as reviewed in this paper, have also shown that each teacher holds a particular belief system comprising a wide range of beliefs about learners, teachers, teaching, learning, schooling, resources, knowledge, and curriculum (Gudmundsdottir & Shulman, 1987; Lovat & Smith, 1995). These beliefs act as a filter through which teachers make their decisions rather than just relying on their pedagogical knowledge or curriculum guidelines (Clark & Peterson, 1986). In fact, these beliefs appear to be cogent enough to either facilitate or slow down educational reform, whichever is the case (Handal & Herrington, 1993, in press). The literature also shows that there are internal and external factors mediating beliefs and practice (Pajares, 1992). This dissonance bears serious implications for the implementation of curricular innovations since teachers’ beliefs may not match the belief system underpinning educational reform. Even if 47
teachers’ beliefs match curricular reform, very often the traditional nature of educational systems make it difficult for teachers to enact their espoused progressive beliefs. In contrast to linear and static approaches to curriculum implementation, modern perspectives look at how teachers make sense of educational innovations in order to re-appraise an ongoing and always flexible process of implementation (Handal & Herrington, 2003). Theoretical Conceptualisations Theoretical conceptualisations of teachers’ mathematical beliefs show that the range of these beliefs can be expressed in multiple dimensions (Kuhs & Ball, 1986; Renne, 1992; Ernest, 1991). Ernest (1991), for example, outlined a developmental sequence of five different mathematics-related belief systems that are hypothesized to be found amongst teachers: authoritarian, utilitarian, mathematics centred, progressive, and socially aware. Ernest’s contribution showed that it is possible to relate these attitudinal representations to conceptions on the theory of mathematics, learning mathematics, teaching mathematics, and assessment in mathematics, as well as identifying beliefs on the aims of mathematics education. According to Ernest, the most important of these categories is the teacher’s philosophy of mathematics, which might vary from absolutist to social-constructivist values. Teachers’ theories of learning and teaching are said to relate to approaches used in class and are fundamental because they define the teacher’s perception of the learner’s role as active or passive, dependent or autonomous, or as receiver or creator of knowledge. Ernest also proposed three main philosophical conceptions of mathematics among teachers. In the instrumentalist view, mathematics is seen as a collection of rules and skills that are to be used for the attainment of a particular goal. Teachers adhering to the Platonist view will maintain that “mathematics is a static but unified body of certain rules” (p. 250) that are to be discovered and are not amenable of personal creation. The problem solving view presents mathematics as a continuous process of inquiry that always remains open to revision. In turn, Kuhs and Ball (1986) characterised three different and dominant conceptions of the ideal teaching and learning of mathematics. The first is the learner-focused view that stresses the learner’s construction of mathematical knowledge through social interaction. The second is the content-focused view with an emphasis on conceptual understanding. The third is the content-focused approach with an emphasis 48
on performance which values performance as the key goal whose attainment depends on the mastery of rules and procedures. Furthermore, Renne (1992) proposed a Purpose of Schooling/Knowledge matrix to conceptualise four different teachers’ conceptions of teaching and learning mathematics. Two groups of teachers are identified in the purpose of schooling category, namely, school-knowledge oriented and childdevelopment oriented. Teachers within the schoolknowledge group believe that teaching is an act of passing information on to others while learning involves the process of reproducing that information. At the same time, school-knowledge oriented teachers place great emphasis on the syllabus and curricular guidelines to guide their instruction. In turn, childdevelopment oriented teachers are more likely to consider children’s needs and characteristics as the primary factors in instructional decision making. The second category in the matrix relates teachers’ beliefs to the way teachers perceive knowledge itself. Schoolknowledge oriented teachers design activities that emphasise acquisition of knowledge in terms of “what” is going to be learned. As such, this type of knowledge is concerned more with rules, procedures, and drill. This type of knowledge is very fragmentary because it does not help the learner relate isolated pieces of knowledge to the whole framework. In contrast, childdevelopment oriented teachers are more concerned with learning of mathematical concepts within an interrelated knowledge structure that is holistic and meaningful. These three different conceptualisations of teachers’ beliefs about the nature and pedagogy of mathematics (Ernest, 1991; Kuhs & Ball, 1986; Renne, 1992) constitute an analytical framework to discuss teachers’ mathematical belief systems. In general, it can be argued that teachers’ belief systems are complex networks of smaller sub-systems operating contextually. The following section attempts to explain the origin of these belief systems within the context of present and past educational environments that appear very traditional and resistant to change. The Cycle of Teachers’ Mathematical Beliefs How do teachers’ mathematical beliefs originate? In part, teachers acquire these beliefs symbiotically from their former mathematics school teachers after sitting and observing classroom lessons for literally thousands of hours throughout their past schooling (Carroll, 1995; Thompson, 1984). This process parallels in many respects the apprenticeship style of Teachers’ Mathematical Beliefs
learning that takes place while learning a trade. Traditionally, tradesmen learn by observing a master doing a particular job (Buchmann, 1987; Lortie, 1975). In the schooling process, students learn not only content-based knowledge but also instructional strategies as well as other dispositions. By the time the aspirant is admitted to a teacher education program, these beliefs about how to teach and learn are deeply embedded in the individual, and very often are reinforced by the traditional nature of some teacher education institutions which may not have positive effects on preservice teachers’ mathematical beliefs (Brown & Rose, 1995; Day, 1996; Foss & Kleinsasser, 1996; Kagan, 1992; McGinnis & Parker, 2001). There is evidence that, in some cases, teacher education programs are so busy concentrating on imparting pedagogical knowledge that little consideration is given to modifying these beliefs (Tillema, 1995). Consequently, teacher education programs might have little effect in producing teachers with beliefs consistent with curriculum innovation and research (Kennedy, 1991). For example, Marland (1994) found that reasons given by inservice teachers regarding their classroom strategies were not related to what was actually taught in their college training. There is also some evidence confirming that teachers’ decision making does not rely solely on their pedagogical knowledge but also on what they believe the subject-matter is and how it should be taught (Brown & Baird, 1993; Laurenson, 1995; Prawat, 1990). These beliefs are also difficult to change (Borko, Flory, & Cumbo, 1993) and very often conflict with educational innovations, threatening educational change (Brown & Rose, 1995; Fullan, 1993). As discussed in the next sections, there are also a number of external factors influencing teachers’ beliefs. The Constraining Nature of Educational Environments The context of school instruction obliges practising elementary and secondary teachers to teach traditional mathematics even when they may hold alternative views about mathematics and about mathematics teaching and learning. Parents and professional colleagues, for example, expect teachers to teach in a traditional way. Teachers are also expected to focus on external examinations, to adhere to a textbook, and to keep a low level of noise and movement in their classrooms. In such environments, even teachers with progressive educational beliefs are forced to compromise and conform to traditional instructional styles (Handal, 2002; Perry, Howard, & Tracey, 1999; Boris Handal
Sosniak, Ethington, & Varelas, 1991). Other accountable factors are ethnic background, social class origins, experience living in other cultures, gender issues, and prior styles of teaching experience (Butt & Raymond, 1989; Raymond, Butt & Towsend, 1991). Thompson (1984) argued that teachers, in the exercise of their practice, and because of the large number and diversity of interactions, tend to develop quick responses to types of episodes, which in time become patterns in their instructional repertoire. McAninch (1993) reviewed a body of literature showing that teachers are very practical in their approach to pedagogical tasks. Jackson’s (1968) interviews revealed that teachers tend to be “confident, subjective, and individualistic in their professional views” (cited by McAninch, 1993, p. 7). In addition, Doyle and Ponder (1977) and Lortie (1975), both cited by McAninch (1993), described “teachers as pragmatic in their decision making…and intuitive in their approach to problem solving” (p. 7). Moreover, teaching is seen as a highly practical and utilitarian profession where teachers quickly label innovations as practical or impractical, depending on whether the teacher considers that the proposal will work for him or her. Success of innovations was also found to be related to a teacher’s personality and teachers were found to emphasise the peculiarities of their classroom over the generalizations of innovations. Nespor (1987) adds that, given the unpredictability and uniqueness of classroom events, teachers have to resort to their own beliefs, particularly in pedagogical situations when formal knowledge is not available, is disconnected, or cannot be retrieved. In Nespor’s words, “When people encounter entangled domains or ill-structured problems, many standard cognitive processing strategies such as schema-abstraction or analytical reduction are no longer viable” (p. 325). This type of situation is characteristic of classroom teaching. In general, teaching is a decision-making based activity in which teachers have to make an interactive decision every two minutes (Brown and Rose, 1995; Clark & Peterson, 1986; Lovat and Smith, 1995). In brief, the teaching job places great external demands on decisions that teachers have to make rapidly, in isolation, and in widely varied circumstances. These demands put teachers in the position of resorting to practicability and intuition as indispensable resources for survival in the profession. These demands in turn favour the development of beliefs about what works and what does not in a classroom. At the same time, it seems that teachers 49
generate their own beliefs about how to teach in their school years and these beliefs are perpetuated in their teaching practice. Thus, educational beliefs are passed on to the students. Teachers’ Instructional Practice If, as the adage says, “teachers teach the way they have been taught” (Frank, 1990, p.12), we need to ask ourselves: what type of mathematics teaching have our and past generations been exposed to? Studies conducted in American mathematics classrooms by Cuban (1984), Mewborn (2001), Sirotnik (1983), and Romberg and Carpenter (1986), Gregg (1995) indicate that most mathematics lessons follow a pattern of whole-class lecturing and “show and tell” style of teaching. Work in small groups is not common and students do not participate actively. Teacher questioning emphasizes right or wrong answers and students are often allocated to passive seatwork. Too much emphasis is given to rote learning, procedures, and facts. It was also found that excess teacher talk dominates in classroom communication and desks usually are arranged to face the teacher’s desk. In sum, this pattern of lessons in American classrooms can be characterised as traditional oriented. Furthermore, the Third International Mathematics and Science Study (TIMSS) identified a similar pattern in Australian classrooms, “one of what might be called ‘traditional approaches’ dominating classroom instruction…particularly in relation to lesson sequencing and types of activities undertaken” (Lokan, Ford, & Greenwood, 1997, p. 231). Based on the above arguments it is possible to suggest that the educational system may act as a vehicle to reproduce traditional mathematical beliefs. Teachers seem to pass on these beliefs in subtle ways in school classrooms. By the time candidates enroll in a teacher education program, these ideas are so solidified and entrenched in their personal philosophy that they will be passed on to their students once the candidates commence their teaching careers, thus carrying on a cycle. The following section attempts to explore the character, intensity, and diversity of these mathematical beliefs as conveyed by schoolteachers. Teachers’ Beliefs about Mathematics and the Learning and Teaching of Mathematics Teachers’ mathematical beliefs are personal and are therefore mental constructs peculiar to each individual (Brown & Rose, 1995). A number of studies have been conducted to obtain “typical” teachers’ mathematical beliefs. Teachers’ mathematical beliefs 50
have been analysed statistically and in many instances judgements were passed on a right-and-wrong criteria by researchers. Although patterns are identifiable within representative samples, these studies have at the same time revealed a broad diversity in the direction and intensity of these beliefs (Carpenter, Fennema, Loef, & Peterson, 1989; Moreira, 1991; Schmidt & Kennedy, 1990). This fact led some researchers to think that these differences could be alternatively interpreted either as stages of a developmental process, individual cognitive differences, or simply due to differences in socio-economic status, educational systems, or cultural environments (Moreira, 1991; Stonewater & Oprea, 1988; Thompson, 1991; Whitman & Morris, 1990). The studies described below show that a large population of teachers still believe that teaching and learning mathematics is more effective in the traditional model, thus suggesting a historical correspondence between teachers’ mathematical beliefs and the teaching practices described in the previous section. What follows is a summary of the main studies conducted to explore mathematical beliefs in preservice and inservice teachers. Mathematical Beliefs of Preservice Teachers A growing body of literature suggests that preservice teachers, that is, student teachers attending teacher education institutions, hold sets of beliefs more traditional than progressive with respect to the teaching of mathematics. Research findings reveal that preservice teachers bring into their education program mental structures overvaluing the role of memorization of rules and procedures in the learning and teaching of school mathematics. For example, Benbow (1993) found that preservice elementary teachers thought of mathematics as a discipline based on rules and procedures to be memorized, and that there is usually one best way to arrive at an answer. Most of the teachers also saw mathematics as dichotomized into “completely right or completely wrong” (p. 10). A similar conservative trend in teachers’ beliefs was reported by Nisbert and Warren (2000), who surveyed 398 primary school teachers with regard to their views on mathematics as a subject, and on teaching and assessing mathematics. Civil (1990) interviewed four prospective elementary teachers and found that they believed that mathematics required neatness and speed, and that there is usually a best way to solve a problem. Frank (1990) surveyed the mathematical beliefs of preservice teachers and found a high level agreement in items such as: (a) “Some people have a Teachers’ Mathematical Beliefs
mathematical mind and some don’t”, (b) “Mathematics requires logic not intuition”, and (c) “You must always know how you got the answer” (p. 11). Moreover, Foss and Kleinsasser (1996, p. 438) surveyed, observed, and interviewed preservice elementary teachers and found that the participants placed great emphasis on practice and memorization. Teachers also were of the opinion that ability in mathematics was innate. Southwell and Khamis (1992) surveyed 71 preservice teachers and found that most participants perceive that mathematics learned in school should be based on memorization of facts and rules. Lappan and Even (1989) and Wood and Floden (1990) report similar findings. Mathematical Beliefs of Inservice Teachers Results from research on inservice teachers show a broader spectrum of responses than with preservice teachers. This is partially the result of more flexible research designs allowing the collection of a broader set of responses in the samples. A number of these studies also show a more varied scope of research questions rather than just simply characterizing teachers’ mathematical beliefs in a dichotomy. The Third International Mathematics and Science Study (TIMSS) (Beaton, et al., 1996), conducted in selected countries around the world, revealed that most teachers believe mathematics is essentially a vehicle to model the real world, that ability in mathematics is innate, and that more than one representation should be used in explaining a mathematical concept. With respect to the emphasis on drill and repetitive practice, teachers around the world did not show a consistent response. Anderson (1997) surveyed and interviewed 25 primary teachers and found that the majority of the participants believe in the value of whole-class discussion, teacher’s modelling, and the use of manipulatives in the classroom. However, it was found that teachers were of the opinion that calculators should not be an important component in teaching mathematics in the primary school. Grossman and Stodolsky (1995) surveyed and interviewed 399 teachers of mathematics, sciences, social studies, and foreign languages. The authors found that mathematics teachers, compared with those of the other subjects, consider their subject highly sequential, static, and have stronger consultation within their faculty for coordinating course content and common exams. The findings also showed that mathematics teachers prefer students to be grouped by prior academic achievement in order to get better benefits from instruction. Schubert (1981), quoted by Brown and Rose (1995), in questioning 123 educators, found that most teachers Boris Handal
believe that pupils learn “in a passive manner by reacting to forces external to them, rather than in an active manner as producer of their own knowledge” (p. 21), a conclusion also supported by Desforges and Cockburn (1987). Finally, Howard, Perry and Lindsay (1997) surveyed 249 secondary mathematics teachers in Sydney, Australia, and found two different patterns of beliefs. The first is identified with the “transmission” profile, that is, a traditional categorization of teaching and learning as the transmission and verification of information in which memorization of rules and procedures is fundamental. This group was larger in number than the constructivist profile, where teachers believe that students are capable enough of constructing their own mathematical knowledge in an atmosphere of negotiation and relevance. The evidence that a large number of inservice teachers hold a diverse collection of mathematical beliefs associated with traditional instruction is also documented in studies conducted by Handal, Bobis, and Grimison (2001), Kifer and Robitaille (1992), Middleton (1992), Perry, Howard, and Conroy (1996), and Perry et al. (1999). Teachers’ Mathematical Beliefs And Instructional Practice Studies on the relationship between pedagogical beliefs and instructional behaviour have reported different degrees of consistency (Frykholm, 1995; Thompson, 1992). While the nature of this relationship seems to be dialectical in nature (Wood et al., 1991) it is not clear whether beliefs influence practice or practice influences beliefs (McGalliard, 1983). It is in fact a complex relationship (Thompson, 1992) where many mediating factors determine the direction and magnitude of the relationship. This section reports a number of studies that have explored the relationship between teachers’ mathematical beliefs and instructional practice. Benbow (1995) conducted an intervention program to deliberately modify the beliefs and instructional practices of 25 preservice mathematics elementary teachers. Findings showed that there was no change in teachers’ mathematical beliefs at the end of the program. However, the researcher stated that instructional behaviour in terms of selection of curriculum content and learning activities, teacher’s role, and teachers’ beliefs on self-efficacy were modified as a result of the program. Lack of pedagogical knowledge and subject-based content were found in some cases to be an obstacle to transfer progressive oriented beliefs into practice. 51
Brown and Rose (1995) conducted an interview study with 10 elementary mathematics teachers in order to determine their theoretical orientations. Teachers’ responses showed a varied range of theories of teaching and learning mathematics. Teachers also said that these orientations influenced their instructional behaviour. The analysis of data revealed that teachers do not implement fully their ideal conceptions of mathematics education because of perceived pressure from parents and school administrators to implement traditional teaching. Other identified mediating factors were the need for more preparation time to satisfy instructional and curricular demands, and the challenges of mixed ability classes. Erickson (1993), in a study with two experienced middle school mathematics teachers, concluded that teachers’ ideal beliefs have a strong influence on their instructional practice. However, obstacles to fully implement their ideals included lack of preparation time and lack of collaboration among peers; size of room; availability of technology, materials, and money; non-supportive administration and parents; need for lengthened class periods; and personal opportunity for growth. Foss and Kleinsasser (1996) studied the behaviour and instructional practice of 20 elementary mathematics preservice teachers. At the end of a onesemester methods course participants had not changed their beliefs about teaching and learning mathematics, which were found to be traditional-oriented and heavily influenced by previous traditional learning experiences in diverse educational settings. Participants’ instructional behaviour replicated or modelled activities learned in the methods course, but not to the extent that reflected an adoption of innovative approaches to teaching and learning mathematics in an articulated and consistent way. In addition, Cooney (1985) studied a beginning mathematics teacher who was committed in belief and in practice to problem solving instruction. The author described the conflict between the teacher’s struggle to teach problem solving and students who preferred a more content-based instruction, a friction that sometimes led to classroom management problems. Perry et al. (1999) studied the beliefs of Australian head secondary mathematics teachers and classroom secondary mathematics teachers as independent samples. Head teachers said that curriculum demands were an obstacle to implementing innovative teaching. In the respondents’ words: We try to make the work relevant but we are constrained by the syllabus. Sometimes, I feel, 52
pressure of the syllabus tends to force us to cut corners with the kids…If I sound cheesed off, it’s just that I may be a disillusioned mathematics teacher. (p. 14)
Raymond (1993) investigated beliefs and practices of six beginning elementary mathematics teachers and found diverse degrees of consistency. Two teachers displayed a high degree of correspondence between belief and practice, two teachers showed a moderate level, while the other two showed a low level. Reasons for the inconsistencies were found to be lack of resources, time limitations, discipline, and pressure to conform to standardized testing. The author concluded that there is a dialectical relationship between beliefs and practice. According to the researcher, teachers’ mathematical beliefs influenced their practice more than their instructional practices influence their mathematical beliefs. The researcher also found that previous school experiences, teachers’ current practice, and, importantly, teacher education courses also influence teachers’ mathematical beliefs. Teachers also identified their own mathematical beliefs, students’ abilities, the particular topic to be taught, the school culture, as well as the mathematics curriculum as factors that influenced their instructional practice. Taylor (1990) attempted to assist a high school teacher to modify his beliefs through a process of conceptual change. However, there were conflicting beliefs, such as the teacher’s belief that he had to teach for constant assessment and for covering the syllabus given that he did not want to jeopardize students’ learning with alternative strategies. Consequently, change in instructional behaviour was restricted. Van Zoest, Jones, and Thornton (1994) interviewed and observed six elementary preservice mathematics teachers participating as students in an intervention program to enhance their teachers’ mathematical beliefs. The authors found that participants acquired beliefs consistent with socioconstructivist views of learning and teaching mathematics, although they were not able to translate these views into practice in the early stages of instructional episodes. The reason for this inconsistency was found in teachers’ lack of pedagogical skill to guide students through the whole problem solving process, time needed to go through a task, teachers’ and students’ tension on how to go about a problem solving situation, and teachers’ concerns about students’ ability to solve the problem. Other studies not showing consistency include Grant (1984) studying secondary mathematics teachers, Kessler (1985) investigating four senior high school Teachers’ Mathematical Beliefs
mathematics teachers, Brosnan, Edwards, and Erickson (1996) researching four middle school mathematics preservice teachers, and Desforges and Cockburn (1987) studying seven experienced mathematics primary school teachers. Thompson (1985) studied two relatively experienced mathematics teachers in their teaching of problem solving and found a high level of consistency between their beliefs and instructional practice. Phillip, Flores, Sowder, and Schapelle (1994) reached the same conclusion while studying four “extraordinary” mathematics teachers. Other studies reporting a strong relationship between teachers’ beliefs and practices have been conducted by McGalliard (1983) investigating senior high school mathematics teachers, and Steinberg, Haymore, and Marks (1985) studying novice teachers. Shirk (1973) working with preservice elementary teachers and Stonewater and Oprea (1988) working with inservice teachers also reported similar consistencies. In general, inconsistencies between teachers’ beliefs and practices are due to constraining forces out of a teachers’ control, such as parental and administrative pressure to follow traditional oriented methods of instruction. Other factors include the traditional oriented mathematical learning style of the students as well as a lack of time and materials. These factors seem to act as major barriers for some teachers in implementing their progressive beliefs, constraints that current approaches in mathematics education do not take into account (Nolder, 1990). Incongruities Between Teachers’ Beliefs And Practice The incongruity between beliefs and practice can also be explained through the agitation and unpredictability of classroom life and the external pressures put on teachers. Thompson (1985) affirmed that these incongruities might be due to the frequency of unexpected occurrences which teachers face in the classroom. The high frequency of these incidents does not permit the teacher to reflect on alternative responses; rather, teachers have time only to react. Jackson (1968) suggested that elementary teachers engage in more than one thousand interactions with students in a single day. Another source of incongruity lies in the personal resolution of conflicting beliefs. Orton (1991) suggested that teachers’ commitment to progressive beliefs is not always a guarantee that these beliefs are going to be translated into practice because sometimes teachers have to compromise their progressive beliefs Boris Handal
for the crude reality of traditional oriented educational environments. For example, a teacher might be motivated to provide rote-learning activities in class when that teacher knows that his or her students will be tested on basic skills in a district proficiency exam. In this case, the teacher might perceive that drill and repetitive practice is the best strategy to attain a temporary goal. Consequent to this strategy, the teacher suspends his or her own progressive beliefs for others that are more central at that particular time. Teacher’s resistance to adopting new approaches in the teaching of mathematics may be part of a defense mechanism that teachers adopt to avoid changes in their own mental structures (Clarke, 1997) because “changing beliefs causes feelings of discomfort, disbelief, distrust, and frustration” (Anderson & Piazza, 1996, p. 53). Orton (1991) stated that it is not easy to change a long-cherished mathematical belief since this belief proved before to be rewarding and useful to the teacher in the performance of his or her professional duties. Furthermore, changing a particular belief implies a re-structuring of the whole network of one’s belief system, a feeling that might cause anxiety and emotional pain (Rokeach, 1968). Concerning teachers’ resistance to change, it has been observed that teachers holding more relativistic orientations to teaching mathematics are more likely to consider and adopt new ideas (Arvold & Albright, 1995). School cultures also influence teachers’ mathematical beliefs (Anderson, 1997). This is particularly true when teachers are found holding beliefs different from the school culture in which they work. For example, a certain school environment might effectively foster values associated with progressive practices and this influence might be stronger than in other schools. In many instances, teachers are caught in a conflict of interest between their “technicalpositivist” and their “constructivist” beliefs and therefore they compromise (Taylor, 1990). Moreover, teachers know that although administrators and supervisors promote reform efforts, professional assessment is in terms of the traditional paradigm and therefore they tend to conform to the status quo to minimize disturbance and professional risk in an ethical-practical way (Anderson & Piazza, 1996; Doyle & Ponder, 1977). Research also shows that teachers may not hold consistent belief systems. Sosniak et al. (1991) analysed mathematical beliefs and self-perceptions of practice of US teachers representing 178 typical eighth grade classes. Based on those responses, the researchers attempted to profile teachers in either a 53
traditional or progressive orientation to the curriculum. However, it was found by statistical analysis that teachers lack a consistent theoretical orientation towards the curriculum. According to the authors, within each teacher’s belief system there are beliefs that appear to be ideologically incompatible with the others. Andrews and Hatch (1999), working mainly with secondary mathematics teachers in the United Kingdom, and Howard et al. (1997) in Australia, reached similar conclusions. Finally, Richardson (1996) adds that in some cases teachers cannot articulate a particular belief because they are unfamiliar with a specific educational innovation. According to Richardson (1996): … it cannot be assumed that all changes in beliefs translate into changes in practices, certainly not practices that may be considered worthwhile. In fact, a given teacher’s belief or conception could support many different practices or no practices at all if the teacher does not know how to develop or enact a practice that meshes with a new belief. (p. 114)
Summary This paper argued that despite many educational reforms, a large number of teachers still perceive mathematics in traditional rather than in progressive terms; that is, as a discipline with a priori rules and procedures, “out-there,” that has to be mechanically discovered rather than constructed. As such, students have to learn mathematics by rote and removed from human experience. The discussion also shows that the relationship between teachers’ mathematical beliefs and their instructional practice is dialectical in nature and is mediated by many conflicting factors. Teachers’ beliefs do influence their instructional practice; however, a precise one-to-one causal relationship cannot be asserted because of the interference of contingencies that are embedded in the school and classroom culture. Even teachers holding progressive beliefs find it difficult to render their ideas into practice due to mediating factors such as the pressure of examinations, administrative demands or policies, students’ and parents’ traditional expectations, as well as the lack of resources, the nature of textbooks, students’ behaviour, demands for covering the syllabus, and supervisory style, among many others. In addition, the teaching profession appears to mould the nature of beliefs because teachers have to make decisions and make meaning of situations quickly, in solitude, with a diversity of subjects, based on empirical knowledge, and under the pressure of external factors. Pedagogical knowledge therefore is 54
not a total predictor of instructional behaviour because beliefs appear to mediate between theory and practice as a powerful interface. Teachers’ mathematical beliefs are seen as self-perpetuating within the atmosphere of a system that promotes progressive teaching but in fact helps in maintaining traditional beliefs and practices. It was also argued that by the time an individual enters a teacher education program, these traditional conceptions are so solidified and entrenched in their personal philosophy that change to alternative beliefs is difficult although not impossible. REFERENCES Anderson, J. (1997). Teachers’ reported use of problem solving teaching strategies in primary mathematics classrooms. In F. Biddulph & K. Carr (Eds.), People in mathematics education. Proceedings of the 20th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 50–57). Rotorua, NZ: MERGA. Anderson, D. S., & Piazza, J. A. (1996). Teaching and learning mathematics in constructivist preservice classrooms. Action in Teacher Education, 18(2), 51–62. Andrews, P., & Hatch, G. (1999). A new look at secondary teachers’ conceptions of mathematics and its teaching. British Educational Research Journal, 25(2), 203–223. Arvold, B., & Albright, M. (1995). Tensions and struggles: Prospective secondary mathematics teachers confronting the unfamiliar. Proceedings of the Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education. (ERIC Document Reproduction Service No. ED 389608.) Beaton, A. E., Mullis, I. V. S., Martin, M. O., Gonzalez, E. J., Kelly, D. L., Smith, T. A. (1996). Mathematics achievement in the middle school years. Boston: Center for the Study of Testing, Evaluation, and Educational Policy, Boston College. Benbow, R. M. (1993). Tracing mathematical beliefs of preservice teachers through integrated content-methods courses. Proceedings of the Annual Conference of the American Educational Research Association. (ERIC Document Reproduction Service No. ED 388638.) Benbow, R. M. (1995). Mathematics beliefs in an “early teaching experience”. Proceedings of the Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education. (ERIC Document Reproduction Service No. ED 391662.) Borko, H., Flory, M., & Cumbo, K. (1993). Teachers' ideas and practices about assessment and instruction: A case study of the effects of alternative assessment in instruction, student learning and accountability practices. Proceedings of the Annual Conference of the American Educational Research Association, Atlanta, Ga. (ERIC Document Reproduction Service No. ED 378226.) Brosnan, P. A., Edwards, T., & Erickson, D. (1996). An exploration of change in teachers’ beliefs and practices during implementation of mathematics standards. Focus on Learning Problems in Mathematics, 18(4), 35–53.
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Teachers’ Mathematical Beliefs
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The Mathematics Educator 2003, Vol. 13, No. 2, 58–59
In Focus… Just Get Out of the Way Adelyn Steele Not too long ago during a typical planning period, I was in my classroom working on lesson plans when in walked my colleague in the math department. As a first year teacher, he is full of questions and ideas and has the ability to see with fresh eyes those situations that some of us just take for granted by now. His question this time had to do with filling out papers for the special education department. After we talked about what he should do and why, he remarked, “You know, at some point I hope to get back to doing what I am supposed to do.” I smiled the sympathetic smile of a paperwork weary comrade and watched him walk out the door. But his statement haunts me and leaves me to wonder: What is it that a teacher is supposed to do? I have asked the question to many people: students, colleagues, principals, parents, and friends; and their answers are startling. Everything from inspire, guide, and rescue; to show up, present facts, and record scores are mentioned. All I know is at the end of every one of these conversations; I feel frustrated, confused, and down right exhausted. So I really don’t know what a teacher is supposed to do (which alarms me slightly as I show up everyday to do it), but maybe the trouble is that I am trying to find a single phrase or idea. A slogan of sorts that could keep me focused and put everything into perspective. Teaching is much too complex for that. There is one idea that keeps coming back to me, however, and that is that the job of a teacher is to set up the task and then get out of the way. Sounds simple, but I assure you it is not. A teacher must either design or find tasks that will allow students to engage in the mathematics. The task must be rich enough to give students something to talk about and wrestle with. It ought to have some significance and build toward an understanding of Adelyn Steele is a mathematics instructor and K-12 mathematics chairperson for Cheney High School in Cheney, KS. She received a BSE in Mathematics from Emporia State University and a Masters in Teaching from Friends University. Adelyn has been named a state finalist in Kansas for the Presidential Award of Excellence in Mathematics and Science Teaching.
mathematics in a way that will be powerful and lasting. I use to think that this was the hard part, but I have since come to understand that selecting the tasks is much easier than deciding what to do with them. Earlier I said that a teacher should get out of the way, and I mean that, but not in the sense of heading to the teacher’s lounge or reading the newspaper in the back of the room. I mean it in the sense of letting the students do as much of it on their own as possible. A teacher should watch the interaction between students and guide what is happening, be on the lookout for evidence of both correct and incorrect observations and understanding, and absorb what is happening in the classroom. Now, most people who have a stake in education seem to say things very similar to what I am saying; yet we disagree at every turn. That is, we disagree with how this is done. The reason, I think, is that we start with different beliefs. See, I believe that understanding and making sense is internal. So if I want that to happen in my students, I need to allow them the time to see that through. If they don’t “get it” in a specific amount of time and I rush in and tell them, then I have robbed them of the opportunity to construct an understanding for themselves. I may have left them with the ability to say back what someone might expect to hear, but they do not own the idea. They are repeating mine. To that end, I need to make the observation that thinking is hard. It can require concentration and awareness to be able to synthesize content. Then a person has to find the words to express what they are thinking. I watch and listen to students, and find that throughout much of their lives people are speaking at them rather than to or with them. If they do not have the opportunity to think and discuss what they are thinking, I believe that those skills are stunted. If, however, students are encouraged to try to find words and are given time to synthesize and articulate then they create within themselves the confidence to wrestle with increasingly complex ideas. I also believe that learning is social. Therefore, my classroom needs to be a social environment where students both talk and listen. They need to respect Just Get Out of the Way
ideas and people and learn to manage themselves in a conversation. (What is this business about having to raise your hand anyway? Where, outside of a classroom, do you see people raising their hands for the opportunity to speak? We owe kids better.) This takes practice and can be messy at the beginning, but as a vehicle to true discourse it is imperative. Core to my beliefs is also the learners’ ability to ask and answer their own questions if given the chance. So often teachers, upon observing and noticing a misconception, jump in to “fix it,” often before a child has the opportunity to really understand that they have that question. I had an algebra teacher who would say to us, “By now you all are thinking….” It was so strange because sometimes we were not thinking whatever he said we were. Once, when this was pointed out to him, he said, “Well, if you are not thinking it now, you should be.” Which brings me to my next belief. People ought to have the right to believe what they want to. This one, ironically, causes all sorts of trouble for me from my colleagues. “So if I want to believe that the sum of 5 and 3 is 7 then that is ok with you?” they have asked. But my point is a real one. So often adults tell students not only what, but also how to think. This creates a dependence that is problematic for years. It is much more powerful to have students defend their own beliefs with data and/or proof. As students do this they develop skills as autonomous learners and profit much more than might be imagined. So what does this have to do with staying out of the way? Well first, if I believe what I say that I do, then my actions will reflect that. So by staying out of the way, I again mean that I let students do as much on their own as possible. I wait. I listen. I probe. I ask
questions. I wait. I listen. I give them an opportunity to think. I am silent. I wait. I listen. I encourage them to test their ideas. I encourage them to talk to each other. I wait. I listen. My intent is to have their thinking on the table for examination by themselves and others, not to dictate what I think. As I listen, I gather information about student understanding (and lack thereof) and look for opportunities for students to build understanding and make sense of the mathematics for themselves. I ask questions much more often than I make statements. I wish to make it clear that I am not passive in the classroom. I am silent much of the time, but that does not equate to passivity. I am actively participating in what students are doing and saying by listening. I will certainly make observations, give relevant information and guide conversations if needed. I just strongly believe that students need very little intervention in their thinking and certainly much less than what they usually encounter. I do not just leave students to falter and stumble, but I only help them up (so to speak) if they demonstrate an inability to do it themselves. The result is not chaos as many might predict. Nor does it take too much time. Certainly there is a trade off in the beginning, but the reward (that of students who think for themselves and articulate and defend their knowledge) far surpasses it in just a few weeks. The beauty of mathematics is that it does make sense and can be explained. Students are just as capable of making sense and building an understanding as anyone else is. The bonus is that if they become confident learners who value their ideas and the ideas of others in addition to the discipline of mathematics at the level they are in, they will have some interest in a future study of mathematics and/or in learning in general.
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The Mathematics Educator (ISSN 1062-9017) is a biannual 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 students, faculty, and alumni of The University of Georgia, as well as the broader mathematics education community. The Mathematics Educator presents a variety of viewpoints within a broad spectrum of issues related to mathematics education. The Mathematics Educator is catalogued in ERIC and 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: • • • • • • • •
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In this Issue, Guest Editorial… What is Mathematics Education For? BRIAN GREER & SWAPNA MUKHOPADHYAY Hidden Assumptions and Unaddressed Questions in Mathematics for All Rhetoric DANNY BERNARD MARTIN The Fourth “R”: Reflection NORENE VAIL LOWERY Impact of a Girls Mathematics and Technology Program on Middle School Girls’ Attitudes Toward Mathematics MELISSA A. DEHAVEN & LYNDA R. WIEST First-Time Teacher-Researchers Use Writing in Middle School Mathematics Instruction DREW K. ISHII Teachers’ Mathematical Beliefs: A Review BORIS HANDAL In Focus… Just Get Out of the Way ADELYN STEELE