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With an eye to the mathematical horizon: Dilemmas of teaching. Paper presented at the annual meeting of the American Educational Research Association
 In J. Brophy (Ed.), Advances in research on teaching
, 1990
"... to this work. 3We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development. It is a bold hypothesis and an essential one in thinking about the nature of a curriculum. No evidence exists to contradict it; consider ..."
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to this work. 3We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development. It is a bold hypothesis and an essential one in thinking about the nature of a curriculum. No evidence exists to contradict it; considerable evidence is being amassed that supports it. (Bruner, 1960, p. 33) The tendrils of this famous passage still wind around current discourse about the improvement of instruction. This paper revisits Bruner's oftquoted assertion that "any subject can be taught effectively in some intellectually honest form. " While my aim is not to suggest that he was wrong, I seek to persuade the reader that figuring out what it might mean to create a practice of teaching that is "intellectually honest " is a project laden with thorny dilemmas and that teachers need to be prepared to face off with the uncertainties inherent in the goal. The new mathematics, science, and history curricula that swept the United States during the 1960s in the wake of Bruner's hypothesis gave us ample evidence that acting on his claim is not easy. This paper takes up the challenge in the particular context of elementary school mathematics: How can and should mathematics as a school subject be connected with mathematics as a discipline? Much current educational discourse centers on the importance of teachers ' subject matter
Research on teaching mathematics: Making subject matter knowledge part of the equation
 in Advances in Research in Teaching, Volume 2
, 1991
"... Subject matter understanding and its role in teaching mathematics are the focus of this paper. Although few would disagree with the assertion that, in order to teach mathematics effectively, teachers must understand mathematics themselves, past efforts to show the relationship of teachers ' mat ..."
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Subject matter understanding and its role in teaching mathematics are the focus of this paper. Although few would disagree with the assertion that, in order to teach mathematics effectively, teachers must understand mathematics themselves, past efforts to show the relationship of teachers ' mathematical knowledge to their teaching of mathematics have been largely unsuccessful. How can this be? My
THE SUBJECT MATTER PREPARATION OF PROSPECTIVE MATHEMATICS TEACHERS: CHALLENGING THE MYTHS
"... I am really worried about teaching something to kids I may not know. Like long divisionI can do itbut I don't know if I could really teach it because I don't know if I really know it or know how to word it. (Cathy, elementary teacher candidate) Teaching the material is no problem. I ha ..."
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I am really worried about teaching something to kids I may not know. Like long divisionI can do itbut I don't know if I could really teach it because I don't know if I really know it or know how to word it. (Cathy, elementary teacher candidate) Teaching the material is no problem. I have had so much math nowI feel very relaxed about algebra and geometry. (Mark, prospective secondary mathematics teacher) I'm not scared that kids will ask me... a computational question that I cannot solve, I'm more worried about answering conceptual questions. Right now, my biggest fearand I'm going to have to confront this on the 3rd of Februaryis what I am going to do if they ask me some kind of question like, &quot;Why are there negative numbers? &quot; (Cindy, prospective secondary mathematics teacher) 2 Cathy, Mark, and Cindy, all preservice teachers, differ in what they think they need to know in order to teach mathematics. While Mark has confidence in the sufficiency of his mathematics knowledge, both Cindy and Cathy suspect that they may come up short when they try to teach. These three teacher candidates represent alternative points of view about the subject matter preparation of teachers. Cathy's viewthat she understands the mathematics herself, but needs to learn to teach itis the basis for traditional formal preservice teacher education. Mark expresses a view that undergirds many of the current proposals to reform teacher education: that people who have majored in mathematics are steeped in the subject matter and have thus acquired the subject matter knowledge needed to teach. Cindy's fear that, although she can do the mathematics, she may not have the kind of mathematical understandings she will need in order to help students learn, is insufficiently shared by those who consider the preparation and certification of teachers. The mathematics knowledge of prospective teachers is the focus of this paper. Despite the fact that subject matter knowledge is logically central to teaching (Buchmann, 1984), it is rarely the object of adequate consideration in preparing or certifying teachers. Three widely held assumptions help to explain this odd state of affairs. First, policymakers and teacher educators seem to assume that topics
Implementing the NCTM Standards: Hopes and hurdles (Issue Paper 922). East
, 1992
"... With increasing pressure to improve American students ' mathematical competence, mathematics educators are trying—again—to change the practices and outcomes of school mathematics. Disappointed by the 1960s efforts at reform (e.g., Sarason, 1971; for an exception, see Romberg, 1990), the communi ..."
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With increasing pressure to improve American students ' mathematical competence, mathematics educators are trying—again—to change the practices and outcomes of school mathematics. Disappointed by the 1960s efforts at reform (e.g., Sarason, 1971; for an exception, see Romberg, 1990), the community watched the sharp shift back to "basics " in the 1970s. Some suspected that this movement was, at least in part, a reactionary swing from the "new math " with its emphasis on abstract mathematical structures. Then, in 1980, the National Council of Teachers of Mathematics (NCTM) published the Agenda for Action, outlining general directions needed to improve mathematics teaching and learning in the 1980s (NCTM, 1980). Although it was widely disseminated, like most documents of its ilk, the Agenda ultimately came to rest on many educators' shelves. A more ambitious move seemed necessary (Cros white, 1990). The two "standards documents " produced by NCTM (1989, 1991) over the past four years represent an unusual step to influence the character and quality of mathematics education.3 One document focuses on curriculum and evaluation, the other on teaching, professional development, and the support and evaluation of teaching. Motivated by a desire to change the way mathematics is taught and learned in school, these documents move the discourse boldly behind the proverbial classroom door and provide new directions
Access to interdisciplinary information: Setting the problem
 Issues in Integrative Studies 7
, 1989
"... Abstract: Identifying and locating interdisciplinary literature, and ideas and information that reside in different disciplines, poses problems for researchers and students. Using electronic means of access, such as online indexes and abstracts and online library catalogs, has provided more flexibil ..."
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Abstract: Identifying and locating interdisciplinary literature, and ideas and information that reside in different disciplines, poses problems for researchers and students. Using electronic means of access, such as online indexes and abstracts and online library catalogs, has provided more flexibility and reduced the amount of time needed for the search process. But scholars continue to question the completeness of the resources for their interdisciplinary work. In part, the problems are due to structures of disciplinary literature and the various forms of access that support current academic and scholarly publications. Scholars can overcome some of the problems with flexible research approaches congruent with the available tools. More importantly, perhaps, groups of interdisciplinary researchers could initiate the development of a taxonomy and language specific to interdisciplinary study and teaching. ACCESS TO INTERDISCIPLINARY information, that is, access to published work that crosses or integrates disciplines, is a multidimensional issue, stemming from both the structure of information in relation to the disciplines out of which it arises, and from scholars’ stances in relation to their research. Contributing to the scholars’
Teaching With The Web: Challenges In A Complex Information Space
"... this paper. 3/1/99 Wallace Proposal Page 28 http://wwwpersonal.umich.edu/~ravenmw/wallaceproposal.pdf ..."
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this paper. 3/1/99 Wallace Proposal Page 28 http://wwwpersonal.umich.edu/~ravenmw/wallaceproposal.pdf
OBSERVING SUBJECT KNOWLEDGE IN PRIMARY MATHEMATICS TEACHING
"... The mathematics subject matter knowledge of primary school teachers has in recent years become a high profile issue in the UK and beyond. This paper describes a videotape study of mathematics lessons prepared and conducted by trainee teachers. The aim was to identify ways in which their subject know ..."
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The mathematics subject matter knowledge of primary school teachers has in recent years become a high profile issue in the UK and beyond. This paper describes a videotape study of mathematics lessons prepared and conducted by trainee teachers. The aim was to identify ways in which their subject knowledge, or the lack of it, was evident in their teaching. We set out a framework which has emerged from the data, in which classroom events and episodes can be viewed as representative of one (or more) of four broad categories. CONTEXT The seminal work of Lee Shulman conceptualises the diversity of the knowledge required for teaching. His seven categories of teacher knowledge include three with an explicit focus on ‘content ’ knowledge: subject matter knowledge, pedagogical content knowledge and curricular knowledge. Shulman (1986) notes that the ways of discussing subject matter knowledge (SMK) will be different for different subject matter areas, but adds to his generic account Schwab’s (1978) notions of substantive knowledge (the key facts, concepts, principles and explanatory frameworks in a discipline) and syntactic knowledge (the nature of enquiry in the field, and how new knowledge is introduced and accepted in that community). For Shulman, pedagogical content knowledge (PCK) consists of “the ways of representing the subject which makes it comprehensible to others…[it] also includes an understanding of what makes the learning of specific topics easy or difficult … ” (Shulman, 1986, p. 9). PCK is particularly difficult to define and characterise, but seems essentially to conceptualise the hitherto missing link between knowing something for oneself and being able to enable others to know it. In 1998, the UK government specified for the first time a curriculum for Initial Teacher Training (ITT) in England (DfEE, 1998), setting out what was deemed to be the “knowledge and understanding of mathematics that trainees need in order to underpin effective teaching of mathematics at primary level”. There is now a growing body of research on prospective primary teachers ’ mathematics subject knowledge, which has undeniably been facilitated by the necessity of some process of audit and remediation within ITT (e.g. Rowland, Martyn, Barber and Heal, 2002; Goulding and
Moving Toward More Authentic Proof Practices in Geometry
"... Various stakeholders in mathematics education have called for increasing the role of reasoning and proving in the school mathematics curriculum. There is some evidence that these recommendations have been taken seriously by mathematics educators and textbook developers. However, if we are truly to r ..."
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Various stakeholders in mathematics education have called for increasing the role of reasoning and proving in the school mathematics curriculum. There is some evidence that these recommendations have been taken seriously by mathematics educators and textbook developers. However, if we are truly to realize this goal, we must pose problems to students that allow them to play a greater role in proving. We offer nine such problems and discuss how using multiple proof representations moves us toward more authentic proof practices in geometry. Over the past few decades, proof has been given increased attention in many countries around the world (see, e.g., Knipping, 2004). This is primarily because proof is considered the basis of mathematical understanding and is essential for developing, establishing, and communicating mathematical
AT
"... This thesis explores the knowledge needed for teaching statistics through investigations at the primary (elementary) school level. Statistics has a relatively short history in the primary school curriculum, compared with mathematics. Recent research in statistics education has prompted a worldwide m ..."
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This thesis explores the knowledge needed for teaching statistics through investigations at the primary (elementary) school level. Statistics has a relatively short history in the primary school curriculum, compared with mathematics. Recent research in statistics education has prompted a worldwide move away from the teaching of statistical skills, towards a broader underpinning of statistical thinking and reasoning. New Zealand’s nationally mandated curriculum reflects this move. Consequently, little is known about the types of knowledge needed to teach statistics effectively. Ideas from two contemporary areas of research, namely teacher content knowledge in relation to mathematics, and statistical thinking, are incorporated into a new framework, for exploring knowledge for teaching statistics. The study’s methodological approach is based on Popper’s philosophy of realism, and the associated logic of learning approach for classroom research. Four primary teachers (in their second year of teaching) planned and taught a sequence of four or five lessons, which were videotaped. Following each lesson, a stimulated recall
WHAT IS THE PHILOSOPHY OF MATHEMATICS EDUCATION?
"... This question (what is the philosophy of mathematics education?) provokes a number of reactions, even before one tries to answer it. Is it a philosophy of mathematics education, or is it the philosophy of mathematics education? Use of the preposition ‘a ’ suggests that what is being offered is one o ..."
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This question (what is the philosophy of mathematics education?) provokes a number of reactions, even before one tries to answer it. Is it a philosophy of mathematics education, or is it the philosophy of mathematics education? Use of the preposition ‘a ’ suggests that what is being offered is one of several such perspectives, practices or areas of study. Use of the definite article ‘the ’ suggests to some the arrogation of definitiveness to the account given. 1 In other words, it is the dominant or otherwise unique account of philosophy of mathematics education. However, an alternative reading is that ‘the ’ refers to a definite area of enquiry, a specific domain, within which one account is offered. So the philosophy of mathematics education need not be a dominant interpretation so much as an area of study, an area of investigation, and hence something with this title can be an exploratory assay into this area. This is what I intend here. Moving beyond the first word, there is the more substantive question of the reference of the term ‘philosophy of mathematics education’. There is a narrow sense that can be applied in interpreting the words ‘philosophy ’ and ‘mathematics education’. The philosophy of some area or activity can be understood as its aims or rationale. Mathematics education understood