Rauwolfia canescens yields another alkaloid which possesses sedative and strong hypotensive activity. This newly isolated ester alkaloid has been termed 11- desmethoxyreserpine. Unlike reserpine, this alkaloid does not yield reserpic acid upon hydrolysis. The acid resulting as a hydrolysis product of 11-desmeth- oxyreserpine has been termed raunormic acid, whose physical properties differ markedly from those of reserpic acid. Acute oral, intraperitoneal, and chronic oral toxicity studies indicate that 11-desmethoxyreserpine exhibits in the test animal a favorable therapeutic index, thus assuring relative safety, warranting further pharmacologic and clinical studies.

By asking the question: “Can computers do mathematics?” this paper investigates the relationship between computers and views of mathematics from both individual and social constructivist perspectives. Although these two perspectives ask many of the same questions, they frame these questions in quite different ways of viewing the interaction between individual, subject matter, culture, and cultural tools (e.g. computers). I argue that whereas social constructivist are more likely to take the position that computers alter the way we do mathematics, individual constructivists would more likely say that computers changes the mathematics that we do. Individual constructivists, by placing mathematics itself within the actions carried out by an individual, provide a theoretical framework that allows for the richness and diversity of student constructions that can expand our understanding of mathematics beyond the bounds of any one particular culture.

Although reform efforts in mathematics education have called for more diverse views of mathematics, there have been few studies of how mathematics is used and takes form in practices outside of mathematics itself. Thus legitimate diverse models have largely been missing in education. This study attempts to broaden our understanding of mathematics by investigating how applied mathematicians and biologists, working together to construct dynamic population models, understand these models within the framework of their perspective practices, that is how these models take on a role as 'boundary objects' between the two practices. By coming to understand how these models function within the practice of biology, the paper suggests that mathematics educators have the opportunity both to reevaluate their own assumptions about modeling and to build an understanding of the dialectic process necessary for these models to develop an epistemological basis that is shared across practices. Investigating this dialectic process is both important and missing in most mathematical classrooms.1

For over 20 years, Ernst von Glasersfeld has eloquently and consistently described a theoretical model of the individual knower. In doing so, he has become one of the best-known contemporary educational theorists, particularly among the mathematics education community.1 In addition, as he states in the preface to his newest book, the name he gave to his approach, “radical constructivism,” has become a catch word among educators. Today it seems everyone is a constructivist and many attach the descriptor radical to their orientation. However, as with any popular movement, there are many interpretations, many critiques, and much passion associated with radical constructivism. Glasersfeld2 sees this volume as an opportunity to tell the whole story, or as he says, “to lay out the main constructivist ideas as I see them” (p. xiii).

Exponential and logarithmic functions are typically presented as formulas with which students learn to associate the rules for exponents/logarithms, a particular algebraic form, and routine algorithms. We present a theoretical argument for an approach to exponentials more closely related to students' constructions. This approach is based on a primitive multiplicative operation labeled ''splitting'' that is not repeated addition. Whereas educators traditionally rely on counting structures to build a number system, we suggest that students need the opportunity to build a number system from splitting structures and their geometric forms. We advocate a ''covariation'' approach to functions that supports a construction of the exponential function based on an isomorphism between splitting and counting structures.

Conventional treatments of functions start by building a rule of correspondence betweenx-values andy-values, typically by creating an equation of the formy=f(x). We call this acorrespondence approach to functions. However, in our work with students we have found that acovariational appraoch is often more powerful, where students working in a problem situation first fill down a table column withx-values, typically by adding 1, then fill down ay-column through an operation they construct within the problem context. Such an approach has the benefit of emphasizing rate-of-change. It also raises the question of what it is that we want to cal ‘rate’ across different functional situations. We make two initial conjectures, first that a rate can be initially understood as aunit per unit comparison and second that a unit is theinvariant relationship between a successor and its predecessor. Based on these conjectures we describe a variety of multiplicative units, then propose three ways of understanding rate of change in relation to exponential functions. Finally we argue that rate is different than ratio and that an integrated understanding of rate is built from multiple concepts.

The interactions of three high school juniors (two females and one male) working together on a series of contextual mathematics problems using a multirepresentational software tool were studied. Focus was on determining how a constructivist model of learning, based on an individual problematic-action-reflection model, can be extended to offer explanatory power for small-group collaborative learning. This extension is constructed by adopting several concepts from the socio-historic or Vygotskian school, including the zone of proximal development, cultural tools, proleptic talk, and appropriation. The subjects worked together during a 10-week secondary mathematJAls course that focused on problem solving with Function Probe. Although constructivist and socio-historic approaches to cognition have, at times, been interpreted as offering opposing viewpointa, it is suggested that there is a potential complementarity, particularly in the area of collaborative peer learning, since researchers in neither area have as yet offered a strong explanatcry model for how students jointly construct mathematical knowledge. Four figures and a 24-item list of references are included. (SLD) *********************************************************************** Reproductions supplied by EDRS are the best that can be made from the original Cocument. *********************************************************************** Understanding Collaborative Learning: Small Group Work on Contextual Problems Using a Multi-Representational Software Tool U.S. DEPARTMENT OF EDUCATION Office of Educational Research and Improvement EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) I/his document has been reproduced as received from the person or organization originating it r Minor changes have been made to improve reproduction quality Points of view or Opinions stated in this document do not necessarily tepresent official OERI positron or policy Erick Smith and Jere Confrey