Jeremy Kilpatrick

Second international handbook of mathematics education

There is much debate within mathematics teacher education over ways in which professional and academic foci could be made to complement each other. On the one hand, teachers’ craft knowledge is emphasized, mainly as this relates to the particular and local level of teaching; on the other hand, the importance of academic subject knowledge cannot be denied. In this chapter the focus will be on how to blend and balance the two through activities in which teachers learn from other teachers, particularly the co-learning of teachers and teacher educators. It will discuss professional relationships, reflective practice, community building, and research in practice. Examples of research-based programs involving lesson study (LS) and the Learner’s Perspective Study (LPS) have moved the relevant research in this area to yet another level, in which theory and practice are combined. Projects such as these and others from diverse parts of the world will be presented and discussed.This chapter seeks to provide an integrating theoretical framework for understanding the somewhat disparate and disconnected literatures on “modelling” and “technology” in mathematics education research. From a cultural–historical activity theory, neo-Vygtoskian perspective, mathematical modelling must be seen as embedded within an indivisible, molar “whole” unit of “activity.” This notion situates “technology”—and mathematics, also—as an essential part or “moment” of the whole activity, alongside other mediational means; thus it can only be fully understood in relation to all the other moments. For instance, we need to understand mathematics and technology in relation to the developmental needs and hence the subjectivity and “personalities” of the learners. But, then, also seeing learning as joint teaching–learning activity implies the necessity of understanding the relation of these also to the teachers, and to the wider institutional and professional and political contexts, invoking curriculum and assessment, pedagogy and teacher development, and so on. Historically, activity has repeatedly fused mathematics and technology, whether in academe or in industry: this provides opportunities, but also problems for mathematics education. We illustrate this perspective through two case studies where the mathematical-technologies are salient (spreadsheets, the number line, and CAS), which implicate some of these wider factors, and which broaden the traditional view of technology in social context.

Mathematics and Cognition: A Research Synthesis by the International Group for the Psychology of Mathematics Education

Forward Contributors Introduction Efrain Fischbein 1. Epistemology and Psychology of Mathematics Education Gerard Vergnaud 2. Psychological Aspects of Learning Early Arithmetic Jacques C. Bergeron and Nicolas Herscovics 3. Language and Mathematics Colette Laborde 4. Psychological Aspects of Learning Geometry Rina Hershkowitz 5. Cognitive Processes Involved in Learning School Algebra, Carolyn Kieran 6. Advanced Mathematical Thinking Tommy Dreyfus 7. Future Perspective for Research in the Psychology of Mathematics Education Nicolas Balacheff References.

Where's the Evidence?

During the last half century, school mathematics in North America has undergone two major waves of attempted reform: the new math movement of the 1950s through the early 1970s and the standards-based movement of the past two decades or so. Although differing sharply in their approach to curriculum content, these reform efforts have shared the aim of making mathematics learning more substantial and engaging for students. The rhetoric surrounding the more recent movement, however, has been much more shrill, the policy differences more sharply drawn, the participants more diverse. The so-called math wars of the 1960s (DeMott, 1962, ch. 9) were largely civil wars. They pitted advocates of rigor and axiomatics against those promoting applied, genetic approaches and were conducted primarily in journal articles and at professional meetings. Today’s warfare ranges outside the profession and has a more strident tone; it is much less civil in both senses of the word. Debates about the new math never became directly implicated in national politics in North America (although they did at times elsewhere in the world). The controversy over recent reform proposals, however, has irresistibly drawn U.S. politicians into the fray—from President Reagan’s plug for one of John Saxon’s mathematics textbooks at a White House reception for school principals in July 1983, to Congress’s 1994 mandate that the Office of Educational Research and Improvement establish expert panels that would identify exemplary and promising educational programs, to Secretary Riley’s 1998 call for a cease-fire in the “math wars,” to last year’s U.S. House Committee on Education and the Workforce’s hearing on the federal role in Grades K‐12 mathematics reform. This high-profile involvement of politicians has helped inflame an atmosphere already heated by melodramatic discourse in the press. In the Wall Street Journal, Lynne Cheney (1997) recounts horror stories of students failing to learn basic skills in the nation’s mathematics classrooms:

Reflection and Recursion

Each age defines education in terms of the meanings it gives to teaching and learning, and those meanings arise in part from the metaphors used to characterise teachers and learners. In the ancient world, one of the defining technologies (Bolter, 1984) was the potter’s wheel. The student’s mind became clay in the hands of the teacher. In the time of Descartes and Leibniz, the defining technology was the mechanical clock. The human being became a sort of clockwork mechanism whose mind either was an immaterial substance separate from the body (Descartes) or was itself a preprogrammed mechanism (Leibniz). The mind has also, at various times, been modelled as a wax tablet, a steam engine and a telephone switchboard.

From the Few to the Many: Historical Perspectives on Who Should Learn Mathematics

Today we take for granted that everybody should be offered the opportunity to learn mathematics. However, it was not until well into the 20th century that “mathematics for all” became an achievable goal. Before then, the geographical location of schools in relation to children’s homes, the availability (or non-availability) of teachers capable of teaching mathematics, parental attitudes to schooling, economic circumstances of families, and social and psychological presuppositions and prejudices about mathematical ability or giftedness, all influenced greatly whether a child might have the opportunity to learn mathematics. Moreover, in many cultures the perceived difference between two social functions of mathematics—its utilitarian function and its capability to sharpen the mind and induce logical thinking—generated mathematics curricula and forms of teaching in local schools which did not meet the needs of some learners. This chapter identifies a historical progression towards the achievement of mathematics for all: from schooling for all, to arithmetic for all, to basic mathematics for all; to secondary mathematics for all; to mathematical modelling for all; and to quantitative literacy for all.

The Chain and the Arrow: From the History of Mathematics Assessment

The problem of assessing the mathematics pupils have learned has inevitably been intertwined with the questions of who should receive additional mathematics instruction and how that instruction should be managed. Scholars appear to have begun the empirical inquiry into these questions from the perspective of psychology during the Renaissance, asking what mental abilities are, how they develop, and how they are differentially disposed across people and across the requirements of various disciplines.

Computers and Curriculum Change in Mathematics

Computers have entered the fabric of modern society so thoroughly that they have become pervasive in everyday affairs. Their entry into schools, however, has been slow, and even slower has been their entry into mathematics classrooms. “While electronic computation has been in the hands of mathematicians for four decades, it has been in the hands of teachers and learners for at most two decades, mostly in the form of time-shared facilities. But the real breakthrough of decentralized and personalized microcomputer-based computing has been widely available for less than one decade” (Kaput 1992, p. 515).

One Field, Many Paths: U. S. Doctoral Programs in Mathematics Education

Background: Mathematics education in the United States: Origins of the field and the development of early graduate programs by E. F. Donoghue Doctoral programs in mathematics education in the U.S.: A status report by R. E. Reys, B. Glasgow, G. A. Ragan, and K. W. Simms Reflections on the match between jobs and doctoral programs in mathematics education by F. Fennell, D. Briars, T. Crites, S. Gay, and H. Tunis International perspectives on doctoral studies in mathematics education by A. J. Bishop Core components: Doctoral programs in mathematics education: Features, options, and challenges by J. T. Fey The research preparation of doctoral students in mathematics education by F. K. Lester, Jr. and T. P. Carpenter The mathematical education of mathematics educators in doctoral programs in mathematics education by J. A. Dossey and G. Lappan Preparation in mathematics education: Is there a basic core for everyone? by N. C. Presmeg and S. Wagner The teaching preparation of mathematics educators in doctoral programs in mathematics education by D. V. Lambdin and J. W. Wilson Discussions on different forms of doctoral dissertations by L. V. Stiff Beyond course experiences: The role of non-course experiences in mathematics education doctoral programs by G. Blume Related issues: Organizing a new doctoral program in mathematics education by C. Thornton, R. H. Hunting, J. M. Shaughnessy, J. T. Sowder, and K. C. Wolff Reorganizing and revamping doctoral programs--Challenges and results by D. B. Aichele, J. Boaler, C. A. Maher, D. Rock, and M. Spikell Recruiting and funding doctoral students by K. C. Wolff The use of distance-learning technology in mathematics education doctoral programs by C. E. Lamb Emerging possibilities for collaborating doctoral programs by R. Lesh, J. A. Crider, and E. Gummer Reactions and reflections: Appropriate preparation of doctoral students: Dilemmas from a small program perspective by J. M. Bay Perspectives from a newcomer on doctoral programs in mathematics education by A. Flores Why I became a doctoral student in mathematics education in the United States by T. Lingefjard Policy--A missing but important element in preparing doctoral students by V. M. Long My doctoral program in mathematics education-A graduate students perspective by G. A. Ragan Ideas for action: Improving U. S. doctoral programs in mathematics education by J. Hiebert, J. Kilpatrick, and M. M. Lindquist References: References by R. E. Reys and J. Kilpatrick Appendices: List of participants by R. E. Reys and J. Kilpatrick Conference agenda by R. E. Reys and J. Kilpatrick.

Commentary on Part I

During the nineteenth century, the study of algebra moved into the secondary school curriculum as colleges and universities began to require it for admission (Kilpatrick and Iszak 2008). Coming after an extensive treatment of arithmetic in the elementary grades, school algebra was commonly introduced formally as a generalization of that arithmetic, with an emphasis on symbol manipulation and equation solving. Given the well-established status of algebra in the secondary curriculum, mathematics educators today confront the question of, in the words of Subramaniam and Banerjee, how “to manage the transition from arithmetic to symbolic algebra.”

Chapter 1 Mathematics curriculum reform in the united states: A historical perspective

Abstract In the United States by the turn of the 20th century, the basic precollege mathematics curriculum of arithmetic, algebra, and geometry was firmly in place. Since then, there have been changes but no substantial reform. The two most significant reform efforts during this century have been the move toward unified and applied mathematics as the century began and the modern mathematics movement of the 1950s and 1960s. Neither of these efforts had its intended effect on the school curriculum, though both left residues. In each case, however, the movement had a profound effect on the mathematics education community, particularly at the post-secondary level. Viewing curriculum reform as a technical rather than a moral and ethical process has led reformers to neglect the basic issues of curriculum discourse.