Gaea Leinhardt

Functions, Graphs, and Graphing: Tasks, Learning, and Teaching

This review of the introductory instructional substance of functions and graphs analyzes research on the interpretation and construction tasks associated with functions and some of their representations: algebraic, tabular, and graphical. The review also analyzes the nature of learning in terms of intuitions and misconceptions, and the plausible approaches to teaching through sequences, explanations, and examples. The topic is significant because of (a) the increased recognition of the organizing power of the concept of functions from middle school mathematics through more advanced topics in high school and college, and (b) the symbolic connections that represent potentials for increased understanding between graphical and algebraic worlds. This is a review of a specific part of the mathematics subject mailer and how it is learned and may be taught; this specificity reflects the issues raised by recent theoretical research concerning how specific context and content contribute to learning and meaning.

Going the Distance With Online Education

This article charts the promissory notes and concerns related to college-level online education as reflected in the educational literature. It is argued that, to appreciate the potential and limitations of online education, we need to trace the issues that bind online education with distance education. The article reviews the history of distance education through the lenses of three historical themes—democratization, liberal education, and educational quality—and charts the current scene of online education in terms of three educational visions that may inform the development of online initiatives: the presentational view, the performance-tutoring view, and the epistemic-engagement view. The article emphasizes the potential contributions of online education to democratization and the advancement of the scholarship of teaching.

Capturing Craft Knowledge in Teaching

This exploration raises some problems and poses some solutions in identifying the craft knowledge of teaching. Craft knowledge, or wisdom of practice, is one important component in the design and validation of new national teacher assessments. The prototype assessment exercises for national board certification are one site in which such craft knowledge has been used. From that experience and others, some guides for inspecting exercises are suggested.

Subject-Matter Knowledge and Elementary Instruction: A Case from Functions and Graphing:

The purpose of the present investigation was to describe the relationship between teachers’ knowledge of mathematics and their instructional practice. An experienced fifth grade teacher was videotaped as he taught a lesson sequence on functions and graphing. In addition, a subject matter knowledge interview and card sort task were conducted with the teacher and a mathematics educator. The results suggest that the teacher’s knowledge of functions and graphing was missing several key mathematical ideas and that it was not organized in a manner to provide easily accessible, cross representational understanding of the domain. These limitations were found to relate to a narrowing of his instruction in three ways: the lack of provision of groundwork for future learning in this area, overemphasis of limited truths, and missed opportunities for fostering meaningful connections between key concepts and representations.

Seeing the Complexity of Standing to the Side: Instructional Dialogues.

In this article, we analyze the complexity of using instructional dialogues in the teaching of mathematics. We trace a 10-lesson unit on functions and their graphs taught by Magdalene Lampert to a 5th-grade classroom. We use this trace to help analyze and systematize the complexity of the classroom discourse. Analysis shows that Lamperts instructional dialogues served 2 purposes: They developed coconstructed instructional explanations of the key mathematical concepts, and they allowed the class to navigate a meaningful path through the relevant mathematics. In creating an instructional explanation, the class as a group flagged the central questions, coordinated and differentiated between the central ideas, and anticipated and made public potential errors. Although misconceptions were often raised as part of the public knowledge space, students individually and collectively resolved these misconceptions publicly through the dialogues. The path through the mathematics was supported through the careful use of agendas, sets of conditions under which an aside was taken from the agendas, and the careful problematizing of the mathematics. Routines, metatalk, and the crafting and maintenance of the intellectual climate played important roles in supporting the instructional dialogues. In the cases of routines and metatalk, we make comparisons to other instances of expert teaching. We also show evidence of student engagement and significant student learning that transcended individual participation patterns. We explore implications for teacher education.

Introduction and Integration of Classroom Routines by Expert Teachers

ABSTRACTSuccessful teachers use the first days of school to establish and rehearse routines which permit instruction to proceed fluidly and efficiently. Routines, shared socially scripted patterns of behavior, serve to reduce the cognitive complexity of the instructional environment. Identified routines were divided into three classes: management, instructional support, and teacher-student exchange. The establishment of routines was examined in the classrooms of six expert teachers, focusing on: (1) the role routines played in the evolving classroom structure; (2) the similarities and differences among teachers in the use of routines; (3) the retention of routines in each of the three identified classes by the teachers; and (4) a detailed look at the most pervasive routines. Teachers were observed to build on their simple routines to form more elaborate strings of action, thus increasing the variety and complexity in the classroom.

Writing from Primary Documents: A Way of Knowing in History.

Developing academic literacy involves learning valued content and rhetoric in a discipline. Within history, writing from primary documents to construct an evidenced interpretation of an issue requires students to transform both background and document knowledge, read and interpret historical documents, and manage discourse synthesis. The authors examine the potential of the Advanced Placement Document-Based Question as constructed and presented by an exemplary teacher to engage students in historical reasoning and writing. The authors analyzed how five students responded to four document-based questions over a year, tracing how organization, document use, and citation language indicate the degree to which writers transformed and integrated information in disciplinary ways. Students moved from knowledge telling (listing period and document content as discrete information bits) to knowledge transformation (integrating content as interpreted evidence for an argument). Students had difficulty learning to handle the complex layers of the task. The authors discuss how instruction might mediate this complexity and promote academic literacy.

Restrictive Educational Settings: Exile or Haven?

This paper reviews the educational and emotional impact of restrictive (isolated) educational settings on children who are in the lowest quartile of achievement. The basic arguments for and against separation of slower students (both those classified as handicapped and those not so classified) are reviewed and some of the more prominent legal cases discussed. Various educational settings are defined and a position is developed as to how setting operates. The main body of the paper examines the results of studies in both special and compensatory education. As a result of the extensive review, it is asserted that the variables which are important for successful student outcomes can occur in most settings, and that for ethical reasons the least restrictive environment is preferred. In summary, it is the issue of effective practices, not the setting, that deserves the attention of educators.

Percent: A Privileged Proportion

Why is percent, a ubiquitous mathematical concept, so hard to learn? This question motivates our review. We argue that asking the question is worthwhile because percent is universal and because it forms a bridge between real-world situations and mathematical concepts of multiplicative structures. The answer involves explaining the long history of the percent concept from its early roots in Babylonian, Indian, and Chinese trading practices and its parallel roots in Greek proportional geometry to its modern multifaceted meanings. The answer also involves specifying what percent is: its meaning (fraction or ratio) and its sense (function or statistic). Finally, the answer involves understanding the privileged language of percent—an extremely concise language that has lost its explicit referents, has misleading additive terminology for multiplicative meanings, and has multiple uses for the preposition of. The answer leads to speculation, in light of previous research, concerning what can be done to teach percent—and other multiplicative mathematical concepts—more effectively.

Mapping Out Geography: An Example of Epistemology and Education:

Geography has recently emerged as a topic of considerable interest among educators, but there is little consensus about either content or pedagogy in the precollegiate geography curriculum. However, geography education is essential if students are to develop a sense of “geographic literacy” and an ability to reason spatially. A major problem is that many teachers who never studied geography are now being asked to teach it. This literature review discusses the epistemology of geography and the rationale for including it in the curriculum. Research about geography learning and teaching is then reviewed. Finally, a previous, unsuccessful attempt to reinstate geography into the high school curriculum is examined in the light of today’s geography education reform efforts. Without an understanding of both the core epistemological themes and concepts of geography and the problems that students face as learners, the five themes currently being proposed for use in the K–12 curriculum are of limited use.