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    Be careful how you interpret the world: It is like that.

    —Erich Heller (Also attributed to Helen Keller)

    A colleague once complained to me that after his daughter had successfully completed her last required mathematics course from a highly regarded univer- sity, she reported to him that as a result of that course she hated mathematics. My colleague was particularly upset because his daughter had enjoyed mathematics up to that point, and he was worried that the negative effects of her last mathematics class might linger long into her future. For many students studying mathe- matics in school, the beliefs or feelings that they carry away about the subject are at least as important as the knowledge they learn of the subject. A 2005 Associ- ated Press poll (AP–AOL News, 2005) showed that nearly 40% of the adults surveyed said that they had hated mathematics in school, and although those polled also acknowledged hating other subjects, twice as many people said that they hated mathematics as said that about any other subject. While students are learning mathematics, they are also learning lessons about what mathematics is, what value it has, how it is learned, who should learn it, and what engagement in mathematical reasoning entails.

    To understand students’ experiences with school mathematics, one must understand a central factor in their experience: mathematics teachers. Two de-

    cades have passed since Lee Shulman (1986) rejected George Bernard Shaw’s infamous statement “He who can, does. He who cannot, teaches” as overly simplis- tic because it was not reflective of the highly special- ized knowledge required of teachers. He introduced the now famous term pedagogical content knowledge to refer to the complex knowledge that lies at the inter- section of content and pedagogy and that teachers must possess to make the curriculum accessible to their students. Shulman ended his landmark article by suggesting that a more apropos saying would be “Those who can, do. Those who understand, teach” (p. 14). Researchers studying teachers’ knowledge, beliefs, and affect related to mathematics teaching and learning are still trying to tease out the relation- ships among these constructs and to determine how teachers’ knowledge, beliefs, and affect relate to their instruction.

    The focus of this chapter is a consideration of what researchers have to say about teachers’ beliefs and affect. A saying by Heller, “Be careful how you interpret the world: It is like that,” indicates that the way one makes sense of his or her world not only de- fines that person for the world but also defines the world for that person. Beliefs might be thought of


    as lenses through which one looks when interpreting the world, and affect might be thought of as a dispo- sition or tendency one takes toward some aspect of his or her world; as such, the beliefs and affect one holds surely affect the way one interacts with his or her world. Although few researchers have examined the relationship between mathematics teachers’ affect and their instruction, the existing research shows that the feelings teachers experienced as learners carry forward to their adult lives, and these feelings are important factors in the ways teachers interpret their mathematical worlds.


    I begin this chapter with a brief review of two chap- ters from the first Handbook of Research on Mathematics Teaching and Learning (Grouws, 1992): one on teach- ers’ beliefs, written by Alba G. Thompson, and a gen- eral chapter on affect, written by Doug McLeod. After summarizing the state of research on mathematics teachers’ beliefs and affect at the time of publication of the first handbook chapter, I review the research conducted on teachers’ beliefs since that time in two sections. In the first section, I consider what beliefs are, how they are measured, what stances are taken on the role of inconsistent beliefs, and how they are changed. In the second section, I review emerging ar- eas of research related to mathematics teachers’ be- liefs by looking to research on teachers’ beliefs about students’ mathematical thinking, teachers’ beliefs about curriculum, teachers’ beliefs about technology, and teachers’ beliefs about gender. I then review the research on teachers’ affect. I begin with a brief review of the relationship between affect and achievement and then summarize studies that address teachers’ affect. Because little research specifically addresses the topic of mathematics teachers’ affect and several researchers have proposed frameworks on students’ affect that might be helpful for considering teachers’ affect, I follow the research on teachers’ affect with a review of some of the student-focused frameworks. Af- ter providing a review of the research on teachers’ be- liefs and affect, I consider researchers who have taken a broader look at the construct of beliefs by consider- ing such constructs as teachers’ orientations, teachers’ per- ceptions, and teacher identity, and I highlight how study- ing identity provides one promising path by which to integrate teachers’ beliefs and affect. I end with some final comments.


    Many of the terms in this chapter are not used in the literature in a uniform way. However, I recognize that readers can become confused about the relationships among terms, and so I have attempted to distill mean- ings that capture distinctions that emerge in usage by researchers, and I list them in Figure 7.1. I call these definitions/descriptions because they are based upon a combination of the literature usage and the diction- ary definitions. These definitions/descriptions are not intended to stand alone as definitions, but in- stead I provide them to support the reader in drawing distinctions among the commonly used meanings of terms in this chapter. Each of these terms is discussed in the chapter, but I provide them at the beginning (see Figure 7.1) so that the reader can refer to them as needed.



    Brief Summary of A. G. Thompson’s 1992 Handbook Chapter on Teachers’ Beliefs and Conceptions

    Twice in her review of the literature on teach- ers’ beliefs and conceptions, A. G. Thompson (1992) noted the importance for researchers studying math- ematics teachers’ beliefs to make explicit to them- selves and to others the perspectives they hold about teaching, learning, and the nature of mathematics, because these perspectives greatly affect researchers’ approaches to and interpretations of their work. In keeping with her own advice, A. G. Thompson stated her stance that researchers must consider the disci- pline of mathematics and the relationship between what a teacher thinks about mathematics and how the teacher teaches. She embraced a view of mathemat- ics “as a kind of mental activity, a social construction involving conjectures, proofs, and refutations, whose results are subject to revolutionary change and whose validity, therefore, must be judged in relation to a so- cial and cultural setting” (p. 127). The conception of mathematics teaching associated with this view of mathematics and reflected in several documents A. G. Thompson cited from the 1980s was “one in which


    students engage in purposeful activities that grow out of problem situations, requiring reasoning and cre- ative thinking, gathering and applying information, discovering, inventing, and communicating ideas, and testing those ideas through critical reflection and argumentation” (p. 128). The role of teachers’ math- ematical conceptions was a recurring theme underly- ing her 1992 review chapter, no surprise to readers fa- miliar with her research (e.g., A. G. Thompson, 1984; A. G. Thompson, Philipp, Thompson, & Boyd, 1994; P. W. Thompson & Thompson, 1994).

    The term belief is so popular in the education lit- erature today that many who write about beliefs do so without defining the term. A. G. Thompson (1992) found, “For the most part, researchers have assumed that readers know what beliefs are” (p. 129). Further, many educators contend that distinguishing between knowledge and belief is unimportant for research, but investigating how, if at all, teachers’ beliefs and knowl- edge affected their experience is important. Although Thompson used both beliefs and conceptions in the title of her chapter, in most of her chapter, she seemed to

    think of beliefs as a subset of conceptions, and her def- inition of conceptions included beliefs. And yet, at times she seemed to use the terms interchangeably. She referred to teachers’ conceptions “as a more general mental structure, encompassing beliefs, meanings, concepts, propositions, rules, mental images, prefer- ences, and the like” (p. 130). When using the term conceptions, Thompson, recognizing the important relationship between knowledge and beliefs, seemed less interested in drawing distinctions between these terms, and she stated, “To look at research on math- ematics teachers’ beliefs and conceptions in isolation from research on mathematics teachers’ knowledge will necessarily result in an incomplete picture” (p. 131). However, early in her chapter she recognized two distinctions between be


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