Teaching University-Level Mathematics Using Mathematica

This paper considers the use of the computer algebra system Mathematica for teaching university-level mathematics subjects. Outlined are basic Mathematica concepts, connected with different mathematics areas: algebra, linear algebra, geometry, calculus and analysis, complex functions, numerical analysis and scientific computing, probability and statistics. The course “Information technologies in mathematics”, which involves the use of Mathematica, is also presented - discussed are the syllabus, aims, approaches and outcomes

Investigations and explorations in the mathematics classroom

In Portugal, since the beginning of the 1990s, problem solving became increasingly identified with mathematical explorations and investigations. A number of research studies have been conducted, focusing on students’ learning, teachers’ classroom practices and teacher education. Currently, this line of work involves studies from primary school to university mathematics. This perspective impacted the mathematics curriculum documents that explicitly recommend teachers to propose mathematics investigations in their classrooms. On national meetings, many teachers report experiences involving students’ doing investigations and indicate to use regularly such tasks in their practice. However, this still appears to be a marginal activity in most mathematics classes, especially when there is pressure for preparation for external examinations (at grades 9 and 12). International assessments such as PISA and national assessments (at grades 4 and 6) emphasize tasks with realistic contexts. They reinforce the view that mathematics tasks must be varied beyond simple computational exercises or intricate abstract problems but they do not support the notion of extended explorations. Future developments will show what paths will emerge from these contradictions between promising research and classroom reports, curriculum orientations, professional experience, and assessment frameworks and instruments

Appealing multimodal languages to access first year university students’ understanding of mathematical concepts in Costa Rica

The current situation regarding the lack of skills and mathematical knowledge that students have when entering the university, has caused that institutions of higher education take certain actions such as the inclusion of courses or content reduction. Most of the measures taken involve curricular changes or partitioning of contents. However, the problem requires also methodological changes that improve students' understanding. Therefore, following the mathematical proficiency and the multimodal approach theories, this qualitative research seeks to use the written languaging exercises that involve the use of natural, symbolic and pictorial languages as a tool to address this situation, promoting the active participation of students to justify and explain their procedures. The aim is to find out student and teachers’ experiences with the languaging exercises. This research was conducted in a Calculus 1 course of the University of Costa Rica, with 33 engineering students and two teachers. The design involves three instruments to collect information: 17 exercises of languaging designed on the topic of derivatives that were applied during the class or as homework during seven weeks, a questionnaire with 18 Likert scale statements and six open ended questions answered by students after the applications of the exercises, and a semi-structured interview for the teachers. The results indicated positive experiences of the participants. They expressed that the languag ing exercises are useful to make learning more meaningful, to identify the different ways in which student’s appropriate knowledge, as well as the misconceptions they have, through the explanations they provide. The exercises also favor, in their opinion, the development of analytical, reasoning, abstract thinking and metacognition skillsUniversidad de Costa RicaUCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de Matemátic

A Curriculum Project on Quadratics Aligned to the Common Core State Standards

This project illustrates a process of designing curriculum for the study of quadratics in Algebra 1, in alignment with the Common Core State Standards in mathematics. The process incorporates the body of research on student conceptions of quadratics and available techniques and technologies for the enhancement of student conceptions. The Algebra 1 course trajectory recommended by the Common Core is analyzed in light of the existing research, unpacked into learning objectives, and restructured to address key conceptual roadblocks synthesized from the research. The resulting curriculum capitalizes on technology, rich problem contexts, and students’ prior knowledge of linear relationships in order to build student concepts of quadratic relationships. The curriculum is sequenced from graphical to symbolic representations, wherein visual and dynamic models provide meaning to symbolic structures and manipulations. A six week instructional calendar and supporting materials are provided to support implementation of this curriculum project

webComputing Service Framework

Presented is webComputing – a general framework of mathematically oriented services including remote access to hardware and software resources for mathematical computations, and web interface to dynamic interactive computations and visualization in a diversity of contexts: mathematical research and engineering, computer-aided mathematical/technical education and distance learning. webComputing builds on the innovative webMathematica technology connecting technical computing system Mathematica to a web server and providing tools for building dynamic and interactive web-interface to Mathematica-based functionality. Discussed are the conception and some of the major components of webComputing service: Scientific Visualization, Domain- Specific Computations, Interactive Education, and Authoring of Interactive Pages

How Ordinary Elimination Became Gaussian Elimination

Newton, in notes that he would rather not have seen published, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method that Euler did not recommend, that Legendre called "ordinary," and that Gauss called "common" - is now named after Gauss: "Gaussian" elimination. Gauss's name became associated with elimination through the adoption, by professional computers, of a specialized notation that Gauss devised for his own least squares calculations. The notation allowed elimination to be viewed as a sequence of arithmetic operations that were repeatedly optimized for hand computing and eventually were described by matrices.Comment: 56 pages, 21 figures, 1 tabl

Analysis of teaching and learning situations in algebra in prospective teacher education

This paper presents a teacher education experiment that was conducted in an algebra course based on an exploratory approach and articulating content and pedagogy. We investigate the contribution of analysing teaching and learning situations, namely student answers and episodes of classroom work, in developing the mathematical and teaching knowledge of prospective primary school teachers. We use a design research methodology to probe the prospective teachers’ development after having participated in an experiment in their third year of a primary education degree program. The results show that the prospective teachers’ understanding of algebra and grasp of how to use different representations and strategies grew considerably. The results also show that their didactical knowledge regarding tasks, classroom organization, attention to students’ reasoning, and teacher’s questions grew as well. The variety of tasks proposed to the prospective teachers during the course was of vital importance to this outcome, as was the opportunity to reflect, work with elements of real practice, and participate in whole class discussions