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In an attempt to promote the development of understanding over rote memorization, writing in mathematics has received increased attention in recent years. In Calculus, the Rule of Three (based on communicating ideas thorough algebraic, graphical and numerical means) has been replaced by the Rule of Four in which writing plays a central role. Educators agree that the benefits of writing include the promotion of understanding, and the initiation of the posing of questions. Writing also helps generate meaning, and helps in the retention of content. In this paper the author evaluates the use of writing for analyzing a problem and its solution. The setting is a reformed differential equations class offered at the Lebanese American University. Unlike a traditional ode course where students are provided with a cookbook of methods for solving differential equations, the emphasis in a reformed ode course is placed on the geometry of the solutions and on an analysis of the outcomes. In many instances, analytically, and the sketch is to be supplemented by an argument justifying it. In addition, various real life problems are modeled and essay questions are asked to analyze the graphs describing these models. Results show that students first reject the idea, but later rate writing as essential. Furthermore, an improvement in the style and content of the writing exercises in usually noticeable at the end of each semester. (Author)

This paper studies the modes of thought that occur during the act of solving problems in mathematics. It examines the two main instantiations of mathematical knowledge, the conceptual and the structural, and their role in the act. It claims that awareness of mathematical structure is the lever that educes mathematical knowledge existing in the mind in response to a problem-solving activity. For didactical purposes it proposes the consideration of mathematical techniques to facilitate the accessing of pertinent knowledge. All the assertions above are substantiated by close examination of some exemplars taken from various mathematical topics, and the presentation of some recent fieldwork results. (Contains 28 references.) (Author)

The undergraduate curriculum in differential equations has undergone important changes in favor of the visual and numerical aspects of the course primarily because of recent technological advances. Yet, research findings that have analyzed students' thinking and understanding in a reformed setting are still lacking. This paper discusses an ongoing developmental research effort to adapt the instructional design perspective of Realistic Mathematics Education (RME) to the teaching and learning of differential equations at Ewha Womans University. The RME theory based on the design heuristic using context problems and modeling was developed for primary school mathematics. However, the analysis of this study indicates that a RME design for a differential equations course can be successfully adapted to the university level. (Contains 37 references.) (Author)

In this paper are described some features of the intensive use of math software, primarily DERIVE, in the context of modeling in an introductory university course in differential equations. Different aspects are detailed: changes in the curriculum that included not only course contents, but also the sequence of introduction to various topics and methodologies. For example, the use of software at the beginning of the course followed a black box strategy, in order to emphasize mathematical modeling and the discussion of theory's central problems. Also covered is the enrichment of discussion about classic problems; using computer algebra systems (CAS) potentialities to demonstrate properties without the necessities of investing excessive time in technical details; avoiding long and boring calculations; design and redesign of materials that attempt to guarantee success of information technology integration; combined use of several software packages; and finally, feedback from students and assessment. In addition, examples are presented to demonstrate the support material's utility for high school calculus. (Contains 19 references.) (Author/YDS)

In this paper are described some features of the intensive use of math software, primarily Derive, in the context of modeling in an introductory university course in differential equations. Different aspects are detailed: changes in the curriculum that includes not only course contents, but also the sequence of introduction to various topics and methodologies. For example, the use of software at the beginning of the course follows a black box strategy, in order to emphasize mathematical modeling and the discussion of theory's central problems. Also covered is the enrichment of discussion about classic problems; using CAS (Computer Algebra Systems) potentialities to demonstrate properties without the necessities of investing excessive time in technical details, avoiding long and boring calculations; design and redesign of materials that attempt to guarantee success of information technology integration; combined use of several software packages, and finally, feedback from students and assessment. In addition, examples are presented to demonstrate the support materials utility for high school calculus. (Contains 5 figures and 19 references.) (Author/PVD)

Computer-aided software has been used intensively in many mathematics courses, especially in computational subjects, to solve initial value and boundary value problems in Partial Differential Equations (PDE). Many software packages were used in student lab assignments such as FORTRAN, PASCAL, MATLAB, MATHEMATICA, and MAPLE in order to accelerate their understanding of concepts and improve their computational skill. This study found that MAPLE was the most effective program in solving, comparing, and visualizing numerical and exact solutions in PDE. Discussion focuses on a computational project to solve, compare, and visualize the solution of heat, wave, and Laplace's equation in PDE. To make a more effective and efficient class, the class was grouped and group members were arranged by mixing them up based on their major skills--theoretical (Pure Mathematics student) and computational (Computational Mathematics student). As a result, it was found to increase the academic atmosphere among the students. Students were found to have new experiences and perceptions about modeling, solving methods, visualization, and real interpretation on PDE problems. (Author/ASK)

Massively parallel programming languages, like StarLogo, provide a rich environment for introducing differential equations to students with an unsophisticated mathematical background. This paper describes the basic software for stimulating and monitoring various population dynamics. Simple differential equations that describe the observed dynamics are introduced. The same environment is then used to explore solutions to the differential equations using Euler's method. Many solutions can be displayed simultaneously and viewed as a flow, which is easier to understand than the more traditional slope fields. All software is written in StarLogo, a massively parallel version of Logo, and can be easily modified, thus permitting students to embark on their own explorations. (Contains 10 references.) (Author/YDS)

This study investigated six students' understandings of and difficulties with qualitative and numerical methods for analyzing differential equations. From an individual cognitive perspective, the obstacles found to influence the development of students' understandings included the function dilemma, the tendency to overgeneralize, interference from informal or intuitive notions, and the complexity involved with graphical interpretations. From a sociocultural perspective, students' understandings were constrained by the use of technology that was disconnected from the learning process, instruction that did not seek out students' explanations, and classroom interactions that implicitly established procedure-based mathematical justifications. Contains 32 references. (Author/NB)

As part of an effort to reform calculus instruction practices, Chattanooga State Technical Community College (CSTCC), in Tennessee, participates in two consortia. One consortium, the Chattanooga Calculus Consortium (CCC), is headed by the University of Tennessee at Chattanooga and involves one other four-year institution and three Chattanooga-area high schools. The CCC was funded by the National Science Foundation (NSF) in 1991 to adapt, implement, and evaluate calculus reform materials being developed at St. Olaf College; inform secondary schools about and involve them in calculus reform movements; prepare computer laboratory manuals; and prepare additional problem sets. The second consortium, the Western Appalachia Calculus Consortium (WACCO), is directed by the University of Kentucky and involves eight other higher-education institutions in three states. Funded by the NSF in 1992, WACCO was designed to share the identification, acquisition, and evaluation of reform approaches and materials through twice annual progress meetings; an Internet-accessible archive of materials; a journal; and workshops with high school educators. Additionally, CSTCC has been working to revise elementary differential equations to move students away from rote memorization to a meaningful understanding of differential equations. Participation in the consortia has benefited CSTCC in the following ways: theoretical and technological assistance has been provided with calculus reform and the college has been allowed input into the reform. The college has shared reform and technology with area high school teachers. (MAB)

Discusses a differential equations course emphasizing conceptual understanding, linear and nonlinear systems, and technology. Presents sample exercises illustrating the difference between this approach and the traditional approach. (MKR)

Reports on the use of technology to enhance the teaching of ordinary differential equations, gives examples of laboratory activities using cooperative learning, and discusses assessment of student learning. MacMath, TI-81 graphing calculators, and Maple were used in the course. (Author/MKR)

Describes undergraduate projects at Harvey Mudd College (California) that use advanced laboratory equipment and procedures normally reserved for graduate students. Examples are given in experimental biology (e.g., digital imaging and DNA analysis), in physics (e.g., using satellites to study earthquake faults), and in mathematics (e.g., teaching differential equations). (LRW)

In an effort to increase faculty use and knowledge of computers, Oakland Community College (OCC), in Michigan, developed a Summer Technology Institute (STI), and a Computer Technology Grants (CTG) project beginning in 1989. The STI involved 3-day forums during summers 1989, 1990, and 1991 to expose faculty to hardware and software applications. The CTG project, initiated in academic year 1990-91, offered 25 microcomputers to faculty on a competitive applications basis. Faculty who were awarded computers, as well as printers and $3,000 for additional hardware and software, were asked to submit written progress reports within 1 year, or to present their work at the STI. After an introduction by George Keith, this report provides the following short papers written by CTG grant recipients: (1) "The Use of Computer in Statistics and Algebra," by Barbara Blass, which reviews a three-semester project to develop a computer assisted course, centered on statistical worksheets and solving algebraic equations using quadratic function graphs; (2) "The Use of Computers in Calculus I," by Gladys Rockind, which describes a program to help students visualize the ideas of limits, tangent lines, derivatives, maximums, minimums, integrals, areas, and volumes and to see the relationships among these areas of calculus; and (3) "The Use of Computers in Calculus IV and Differential Equations," by Keith Shuert, which describes the use of computers to display and rotate multivariate surfaces, to graph the fields and solution curves of differential equations, and to conduct numerical simulations. The bulk of the report consists of sample materials and worksheets from the course taught by Blass. (PAA)