Book Review

On the Connections Between General Education Theories and

Theories in Mathematics Education

Theories of Mathematics Education: Seeking New Frontiers. Bharath Sriraman and Lyn English (Eds.) (2010). Heidelberg, Germany: Springer, 668 pp. ISBN 978-3-642-00741-5 (hb) $129, (e-book) $103.20.

Reviewed by Roza Leikin, University of Haifa, and Rina Zazkis, Simon Fraser University

Theories of Mathematics Education: Seeking New Frontiers is the first book in the series titled “Advances in Mathematics Education.” The book contains 20 chapters, 16 prefaces, and 23 commentaries, and, according to the editors,

synergizes the efforts of numerous groups across the globe in the ongoing debates on theory development in mathematics education. The book presents a good case that theory development is indeed progressing . . . and our field has indeed consolidated and synthesized previous work and moved forward in unimagined and productive ways. (p. x)

Our review focuses on the ways in which, according to the book, consolidation and synthesis of previous work lead to development of theory. We focus our atten-tion on the relationships between general educational theories and theories in mathematics education (ME), as well as on the mutual relationships between theory and research in ME (RME). We first outline the position of the contributors to this book on the question: “Is there such a thing as a theory of mathematics education?” Then we address the issue of borrowing wisdom from other fields and argue that ME theories and models are ready to inform the more general educa-tional community. We turn to the interconnection among mathematics, psychology, and social sciences within the field of ME and consider some examples from the book. We examine and support the argument of the book’s editors that networking and consolidation of the efforts of different researchers can advance theoretical development in the field of ME. Finally, we suggest what different readers of the book can learn, or miss, when reading the book.

THEORY OF MATHEMATICS EDUCATION—IS THERE SUCH A THING?

Kilpatrick (in the preface to part 1) points out the conflict between views on the theory of ME as versioned by Silver and Herbst (2007) and those by Sriraman and English (2010). According to Kilpatrick, Silver and Herbst (2007) argue that the

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development of a “grand theory of mathematics education” (p. 4) is “not simply attainable but desirable [emphasis added] for organizing the field” (p. 4). Sriraman and English, on the contrary, “[point] out that creating a grand theory would be difficult if not impossible [emphasis added]” (p. 4), because it requires “extracting mathematics teaching and learning from the social and cultural context” (p. 4). Stemming from such a position, Sriraman and English assume that this is a question “for ongoing debate” (p. 4). Within this debate we argue that the extraction is not necessary; rather, integration of ME in general education research and viewing ME not only as informed by more general theories but also as informing general theo-ries can advance the construction of a theory of mathematics education. Along with the position of Sriraman and English, Kilpatrick argues:

To say that something is a theory of mathematics education—rather than, say, an approach, theoretical framework, theoretical perspective, or model—is to make an exceedingly strong claim. I would not award theory-of-mathematics-education status to any of the potential contenders cited either by Silver and Herbst (2007) or by Sriraman and English. (p. 4)

Several contributors to this book (e.g., Kilpatrick, Lester, Harel, Lerman, Pegg and Tall, Dahl, Wedege, and Hurford) acknowledge the distinction between theo-ries, approaches, theoretical frameworks, and models. Although there is common agreement that mathematics education needs to connect theory and practice, theo-rizing about teaching and learning leads to the design of theoretical perspectives, models, and approaches to teaching and learning mathematics. The models and theoretical perspectives inform practice, are informed by practice, and are directed at solving practical problems in ME. RME, in turn, has to be based on a clear theoretical framework that determines research methodology and advances under-standing of theoretical assumptions and hypotheses (see Figure 1).

Although distinctions between theories and models are not clear enough, we are

Figure 1. Theories/models of and in mathematics education.

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at times uncertain about the classification of particular contributions to this book as theories of ME. For example, Sinclair presents novel and insightful ideas about “knowing more than we can tell,” inviting a researcher’s look beyond the obvious. When reading this inspiring chapter we do not doubt the significance of this contri-bution in providing new insights to mathematical understanding, thinking, and learning. However, we wonder whether she has designed a theory or a model, and under what definition her views can be considered to be a “theory.” Would naming her chapter A Theory of Covert Knowledge turn her ideas into a theory?

Lester discusses theoretical, conceptual, and philosophical foundations of RME. He stresses that searching for “what works” in ME narrows researchers’ perspective and distances the researchers from philosophical questions and from the develop-ment of theory in the field. Lester argues that a theoretical framework is a necessary condition for conducting research that leads to a deep understanding of the phenomena under investigation. He draws a distinction between practical frame-works, which are not informed by formal theories, and conceptual frameworks, which are structures of justifications. Justifications in RME, as opposed to expla-nations provided by the researchers of the observed phenomena, inform research at the metalevel, illuminate reasons for taking a particular research approach to exploration of a phenomenon, and provide a particular type of explanation of the research findings. Lester argues that the development of theory is “absolutely essential” (p. 74) to allow for significant progress in RME.

EXAMPLES OF APPLICATION—FROM THEORY TO RESEARCH

In the book we find several important illustrations of the implementation of theories in research. For example, Harel reintroduces a research-based framework that he developed over the past decade, DNR-based instruction (in short, DNR), which relies on the centrality of three principles: Duality, Necessity, and Repeated Reasoning. He explains how this framework evolved from his previous well-known work on proof schemes and presents an analysis of an algebra problem-solving lesson, in terms of this framework. Although concerns are raised with respect to the feasibility of implementation of DNR-based instruction in today’s school climate (Sriraman), we see the main identifying feature in Harel’s work as attention to mathematics. He explicitly mentions mathematics (or mathematical knowledge) as the first premise on which DNR is based. Furthermore, although explicit “atten-tion to mathematical content is peripheral [we add—absent] in many current frameworks and studies in mathematics education” (p. 343), Harel’s work is one of several much-needed counterexamples of this tendency.

Another example can be seen in the chapter by Törner, Rolka, Rösken, and Sriraman, who use Schoenfeld’s (1998) teaching-in-context theory to analyze the complexity of interaction in the mathematics classroom. They elaborate on Schoenfeld’s knowledge-goals-beliefs (KGB) structure in describing the teacher’s decision making during a well-planned lesson that did not turn out as expected. As such, they refine the theory by elaborating upon the interrelationship among

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different “belief bundles.” Furthermore, they extend the scope of applicability of Schoenfeld’s theory—developed to model a series of tutoring and teaching episodes—to a new setting, a Grade 8 classroom in Germany.

Harel stresses that one of the identified weaknesses in RME is “widespread misunderstanding among researchers of what it means to adopt a theoretical or conceptual stance toward one’s work” (p. 343). In spite of the strong and insightful examples this book delivers to its readers, it is our position that the book highlights the problematics outlined by Harel and calls for resolving the issue.

BORROWING THEORIES FROM OTHER FIELDS AND APPLYING THEM TO MATHEMATICS EDUCATION

Mathematics education research, as well as theories in ME, borrows from cogni-tive and educational psychology (as related to learning and teaching), sociology, anthropology, and, lately, neuroscience. The question of the extent to which ME theory stands alone is open. The paradigms in RME change over time along with changing approaches in psychology, sociology, and other sciences (e.g., behav-iorism vs. constructivism).

In this direction Ernest, in the chapter “Reflections on Theories of Learning,” tracks the development of constructivism as a learning philosophy. He discusses four modifications of constructivism—“simple” constructivism, radical construc-tivism, enactivism, and social constructivism—as exemplifying the development of modern conceptions associated with ME and, probably, with education in general. He demonstrates that the four modifications are mutually connected but are based on different theoretical foundations and their combination (e.g., Piaget, 1972; Von Glasersfeld, 1989; Vygotsky, 1978; Wittgenstein, 1953).

Whereas Ernest himself does not talk about the theory of ME, but rather about the philosophy of learning in ME, Goodchild in his commentary to Ernest’s chapter calls these philosophies “theories.” Following this commentary Ernest (in Commentary 2 on “Reflections on Theories of Learning”) explains that philoso-phies should be testable enough to be theories. He admits, however, that following the common usage he applies the use of the term learning theory to different forms of constructivism. Ernest continues the discussion of the different bases for the theories in ME by referring to four orientations of learning suggested by Wegerif (2002). Based on Wegerif, Ernest points out that four orientations (behaviorist, cognitivist/constructivist, humanist, and participatory) were initiated by different theorists who differ in their views of the learning process, locus of learning, purpose of education, and educators’ roles.

One of the interesting examples of advances in the theory of ME is based on the combination of the latest developments in biological sciences, technology, and cognitive neurosciences. Campbell elegantly calls his chapter “Embodied Minds and Dancing Brains: New Opportunities for Research in Mathematics Education” to outline the connection between the broadly accepted theoretical framework of embodied cognition (e.g., Lakoff & Johnson, 1999) and the relatively new direction

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of brain research in ME. Campbell considers brain research as a continuation of the ideas of embodiment, that is, bodies are prior to ideas (Lakoff & Johnson, 1999), and the brain is a part of our body. He discusses why researchers in ME should care about brain research and explains its complexity, as reflected in the very slow pace of development in this research branch. The chapter points out the new tools and new questions that can be asked when performing brain research in ME. However, this intriguing part of the book does not provide detailed information about neurosci-ence techniques and approaches that might be useful in ME research. For example, ERPs (Event Related Potentials) are electrophysiological measures reflecting changes in electrical activity of the central nervous system related to external stimuli or cognitive processes occurring in the brain. ERP technique has useful applications in language-related research (see Beaton, 2004, for a review) and recently was adapted for the study of mathematical processing (O’Boyle, 2005). Even so, while in the language domain a great number of electrophysiological studies have been performed, the situation is different in the field of mathematics. Conducted mostly in the 1990s, these studies were very diverse from a methodological point of view. They used different technical tools (i.e., PET, EEG ERP and fMRI), different age groups, and different (but mostly easy) mathematical and nonmathematical tasks (see O’Boyle, 2000, 2005, for a review). Yet, little has been discovered through brain research about mathematical understanding and learning, and it appears that ME researchers are just beginning to envision the possibilities of brain research in ME.

Another example of borrowing theories and approaches from other fields for theoretical frameworks in ME is analyzed by Lesh and Sriraman by suggesting reconceptualization of ME as a design science. The authors suggest that ME research can be enriched and enhanced if considered “akin to engineering and other emerging interdisciplinary fields which involve the interaction of ‘subjects,’ conceptual systems and technology influenced by social constraints and affor-dances” (p. 123). The analogy is made to research in science and engineering and acknowledges education as a complex system. Lesh and Sriraman suggest that such a framework can help researchers ask more meaningful research questions through focusing on structures rather than on distinct pieces of information. By considering ME as a complex system, Lesh and Sriraman doubt the possibility of designing a global theory of ME; rather, they argue for a variety of theories and consider the power of models as compared to theories in education. They analyze integration of theories and methodologies through models and the modeling process as in engi-neering and sciences. Models—in their view—contribute to theories, whereas theories can be included in models and the modeling process when the problems are situated outside the theories.

RME: BETWEEN MATHEMATICS, PSYCHOLOGY, AND SOCIAL SCIENCES

Sriraman and English argue that ME was long considered as an intertwinement between mathematics (particular contents) and psychology (cognition, learning,

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and pedagogy), and lately has been influenced heavily by social, cultural, and political dimensions of education. Harel, in his commentary to Lester’s chapter, discusses the role of mathematical context in RME. In his view:

The goals of MER are to understand fundamental problems concerning the learning and teaching of mathematics and to utilize this understanding to investigate existing products and develop new ones that would potentially advance the quality of mathe-matics education. (p. 91)

As quality of ME is expressed in students’ mathematical understanding and students’ ways of thinking developed through instructional process, we argue the centrality of the mathematical context for RME and theory design.

When reading this book we were looking for ways in which the authors use mathematical content to describe, discuss, critique, and compare theories and models in mathematics education. We observe that very little attention is devoted to concrete mathematics topics and that very few mathematical concepts are involved in the chapters. Based on this observation we ask: What is specific to theories in mathematics education as compared to theories in other-subjects educa-tion?

At first glance, the fact that very few references are made in the book’s chapters to specific mathematical concepts or topics can be seen as a weakness in the discus-sion of theories of ME. On the other hand, this “lack of concreteness” can be considered as an indication of the generality of the constructed theoretical frame-works; thus their use is probably broader than is assumed by the authors. We wonder whether theories of ME can inform general theories and to what extent theories in ME are specific to mathematics. For example, Brousseau’s (1997) theory of didac-tical situations possibly can be considered as a more general construct, while the concepts of didactical situation, didactical contract, and devolution of a good task can be helpful in teaching and learning sciences, languages, history, and literature.

In our search in the book for the applications of mathematical content to theories of ME, we find several analogies that the authors draw between ME theories and mathematics as a discipline. For example, Lesh and Sriraman, when discussing mutual relationships between models and theories in ME, use the following example to create an analogy with mathematics:

A Cartesian Coordinate System may be referred to as a model of space—even though a Cartesian Coordinate System may be so large that it seems to be more like a language for creating models rather than being a single model in itself . . . Modeling students [sic] modeling is the study of a complex living system with layers of emerging ideas, sense making and a continuous evolution of knowledge, which suggests we adopt a phylogenetic approach to modeling the growth of knowledge and learning. (p. 145)

Another example of such a connection can be seen in Lerman’s discussion of hierarchical and vertical or horizontal types of discourses within theories in ME (cf. Bernstein, 2000). “Knowledge discourses are described as hierarchical where knowledge in the field is a process of gradual distancing, or abstraction, from everyday concepts” (p. 100). As such, mathematics as a discipline represents a hierarchical type of knowledge discourse. Scientific discourse is said to be vertical,

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meaning that previous theories are integrated by new ones. When new discourses develop alongside the previous ones, they can be considered to have a horizontal structure. Natural sciences, argues Lerman, are of a vertical structure, whereas mathematics education is of a horizontal one. He explains that, analogously to the case of non-Euclidean geometries, inclusion of Vygotsky’s works in RME did not lead to the replacement of Piaget’s theory or to the incorporation of Piaget’s theory into an expanded theory. “Our study suggests that there is a growing range of theories being used in our field . . . Theories do not disappear, and the number and range of theories is proliferating” (p. 107).

PLURALITY AND NETWORKING

“Is plurality a problem?” is the question posed by Lerman in the title of his chapter. A short answer derived from reading the book (and not necessarily Lerman’s position) appears to be no: Without plurality of theories, this publication would not exist. However, a long answer suggested by Lerman and others identifies several problematics that can be attributed to plurality. “Joining together” different theories is often done in an unsatisfactory fashion. So the problem is not in plurality itself, but rather in the ways that the theories are devised and applied. The prolif-erating number of theories suggests a horizontal rather than hierarchical growth, which means there is insufficient reliance on prior work and no accumulation thereof. This—as Lerman delicately suggests—may be connected to researchers’ need to create self-identity. Furthermore, Lesh and Sriraman identify a “poor record of accumulation” (p. 139) as the main problem of RME, not only with respect to the development of theories, in particular pointing to research on problem solving. Dahl adds to this that there is insufficient testing of theories and, in such, no falsi-fication, where falsification may be a partial means of reducing plurality.

“A series of rival sub-areas with little dialogue” is how ME can be described, according to Jablonka and Bergsten (p. 114). However, although subareas are essential for any expanding field, and a dialogue is something for which to strive, we suggest that “rivalry” is not necessary. Although some world-views appear incompatible (Lerman), some mathematics educators prefer the notion of the complementarity or incommensurability of theories (Kuhn, 1996), that is, having no common measure or no common grounds for comparison. (Irrational numbers come to mind when thinking of incommensurable measures; still, those can appear infinitesimally close to each other.)

Simon (2009) discusses the advantages of considering the same classroom situ-ation through different theoretical lenses, where “each lens affords a different view of the same situation” (p. 484). This basic idea is elaborated upon in several chap-ters. Bikner-Ahsbahs and Prediger describe a variety of ways in which theories can be “networked”; that is, either integrated, contrasted, or combined, and how new theories may emerge from integrating various perspectives. Pegg and Tall provide an extended discussion of comparing and contrasting several theories related to concept construction.

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Jungwirth provides an explicit example of networking—by synthesizing and coordinating—in analyzing an instructional episode with the “effective combina-tion” (p. 519) of two theories: the microsociological theory and the linguistic activity theory. Gellert criticizes the notion of “theorizing as bricolage” (Cobb, 2007; Lester, 2005), where “the bricoleur takes whatever tool is at hand; the researcher constructs the optimal tool for the very research purpose” (p. 539). He illustrates construction of such an optimal tool by presenting integration of semiotic and structuralistic interpretations of the same instructional interaction. Both Gellert and Jungwirth elaborate upon the benefits of such integration. This linkage among theories implies more than a “peaceful coexistence” of theories that are “incom-mensurable” (Sfard, 1998); we suggest that it can be described metaphorically as a symbiosis, that is, a relationship advantageous to both organisms.

SEEKING FURTHER DISCUSSION

If you wish to get a comprehensive overview of theories that have been used in RME, look elsewhere. If you are looking for multiple examples of novel theories and their usage in RME, look elsewhere. This is, of course, if novel is interpreted as not published previously. However, if you are interested and ready to join, as a listener, an interesting and informative conversation among the leaders in the field of ME, to follow their contemplations and become exposed to their arguments and disagreements, then this book will provide you with plenty. You are likely to be inspired by some of the ideas, but—a warning—you may also become confused.

A reader will find in the book a compelling argument for the need for a theory or a theoretical perspective; an insightful conversation among researchers—including veterans—shaping the landscape of ME; a variety of arguments and ideas related to the multitude of theories, including a variety of ways of linking several theories; a discussion on problematics with current usage of theories in research—these include lack of accumulation of knowledge as well as insufficient attention devoted to the development in other fields; examples of the application of a theory in data analysis; and a very long list of topics that influence ME beyond teaching and learning and that broaden the scope of the field. This is only a partial list. A comprehensive review of the work of this scope—about 700 pages, 20 chapters, 16 prefaces, and 23 commentaries—would be impossible within the space limits of this review. We have chosen to focus only on several selected issues and as such to refer only to several authors. Our apologies to those that are not featured or mentioned herein: In no way do our choices reflect our views on the importance of their contributions or our personal preferences. Different perspectives on the book and additional discussion of the contributions of different authors can be found in, for example, Ely (2010), Jankvist (in press), and Schoenfeld (2010).

We find the organization of the book rather novel, in that each chapter constitutes its own part with “prologue” and “epilogue.” However, this structure has its prob-lems. In many places we wondered whether the introduction and reaction could be interchanged. We also wondered—and hoped that editors would have foreseen and

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addressed the question—why some chapters contain both a preface and commen-tary, whereas others do not, and why some chapters have several commentaries. Furthermore, because each chapter is presented as its own part, we initially felt that more high-level organization is lacking. Such an organizational scheme would present “parts” either according to major philosophical perspectives or by catego-rization, such as theories of learning, theories of teaching, or theories of knowledge. However, a more detailed look at the chapters convinced us that the intertwined nature of the issues that the chapters address would make such a separation undesir-able, if not impossible.

The book amplifies the RME community’s need for some kind of “handbook” of theories and their usage. If such a volume were created, it would present a multi-tude of interpretations of the concept of theory and take a stand on how this should be interpreted in ME, supporting an argument for the importance of consistency, not in a theory itself but in interpretation of what a theory is. Then, it would present a comprehensive overview of existing “popular” frameworks and models, situating those within major “global” perspectives and articulating in what way local theories fit in within major ones. As an alternative to focusing on “global” theoretical perspectives, the structure may separate theories of knowledge, theories of learning, and theories of teaching. Furthermore, it would exemplify theory application by providing summaries of research studies. Those could include implementation of “single” frameworks or perspectives, as well as possible linkages and networking. Such a collection would require a major collaborative effort, but the enthusiasm and professionalism that Sriraman and English demonstrate in creating this volume suggest that they are ready for the challenge.

So, aligning the contents of the book with its title, it might be useful to think of theories in (or used in) ME or in psychology in ME. It might also be useful to emphasize the word seeking in seeking new frontiers, as the book does not presume to present total novelty, but to reach toward what could be useful considerations in future research and theory design.

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