Mathematics without Lectures: Small-Group Learning at New York University

This article describes an effort to introduce small-group learning into the mathematics curriculum for the non-specialist at New York University. Starting in spring 1999, students were offered the choice of fulfilling their mathematics requirement in a small-group environment that included no formal lectures. The goal of these groups is to make the transition from relatively inactive, even passive, lectures to an experience that actively engages students in the process of doing mathematics. Contact time was restricted to two weekly classes run by a graduate student and was limited to enrollments of 15-16 students. The course is a small-group version of one that has been offered regularly since 1995, with a format that includes two traditional large lectures and one 100-minute workshop each week. Students in the College of Arts and Science and in the School of Education took the course, and the latter group included future K-12 teachers. Instructors for the small-group sections come from the graduate level Mathematics Education Group in the School of Education and the Mathematics Department in the Graduate School of Arts and Science

On the integration of digital technologies into mathematics classrooms

Trouche‘s (2003) presentation at the Third Computer Algebra in Mathematics Education Symposium focused on the notions of instrumental genesis and of orchestration: the former concerning the mutual transformation of learner and artefact in the course of constructing knowledge with technology; the latter concerning the problem of integrating technology into classroom practice. At the Symposium, there was considerable discussion of the idea of situated abstraction, which the current authors have been developing over the last decade. In this paper, we summarise the theory of instrumental genesis and attempt to link it with situated abstraction. We then seek to broaden Trouche‘s discussion of orchestration to elaborate the role of artefacts in the process, and describe how the notion of situated abstraction could be used to make sense of the evolving mathematical knowledge of a community as well as an individual. We conclude by elaborating the ways in which technological artefacts can provide shared means of mathematical expression, and discuss the need to recognise the diversity of student‘s emergent meanings for mathematics, and the legitimacy of mathematical expression that may be initially divergent from institutionalised mathematics

Using the Arts to Develop a Pedagogy of Creativity, Innovation, and Risk-Taking (CIRT)

This paper considers the complex and somewhat nebulous term “creativity”, exploring the ways in which the pedagogical phenomenon we call “CIRT” (an acronym) can enrich classroom approaches so as to enhance Creativity, boost Innovation, and encourage Risk-Taking. In addition, we review elements that impact the creative process and explore concepts of freedom, as well as the constraints and parameters of creativity. In our role as teacher educators, we explore the connection between teaching and creativity by outlining three key examples of approaches that utilize the CIRT framework including: synesthesia, imagination, and audiation activities.  

The role of pedagogical tools in active learning: a case for sense-making

Evidence from the research literature indicates that both audience response systems (ARS) and guided inquiry worksheets (GIW) can lead to greater student engagement, learning, and equity in the STEM classroom. We compare the use of these two tools in large enrollment STEM courses delivered in different contexts, one in biology and one in engineering. The instructors studied utilized each of the active learning tools differently. In the biology course, ARS questions were used mainly to check in with students and assess if they were correctly interpreting and understanding worksheet questions. The engineering course presented ARS questions that afforded students the opportunity to apply learned concepts to new scenarios towards improving students conceptual understanding. In the biology course, the GIWs were primarily used in stand-alone activities, and most of the information necessary for students to answer the questions was contained within the worksheet in a context that aligned with a disciplinary model. In the engineering course, the instructor intended for students to reference their lecture notes and rely on their conceptual knowledge of fundamental principles from the previous ARS class session in order to successfully answer the GIW questions. However, while their specific implementation structures and practices differed, both instructors used these tools to build towards the same basic disciplinary thinking and sense-making processes of conceptual reasoning, quantitative reasoning, and metacognitive thinking.Comment: 20 pages, 5 figure

An analysis of engineering educators’ understanding of complementary studies courses using the repertory grid technique

Accreditation bodies such as the Engineering Council of South Africa and the Canadian Engineering Accreditation Board have a group of courses that fall under the umbrella of Complementary Studies. This term is used to describe a set of engineering courses that include knowledge areas other than the more common mathematical sciences, natural sciences, engineering sciences, design and synthesis, and workintegrated learning. Studies have shown that engineering educators sometimes view these courses negatively. They are seen as distracting the focus of the students on the so-called technical courses, which the educators feel are more important. This paper reports on a research study that explored the way that engineering educators make sense of complementary studies courses within an industrial engineering curriculum. The repertory grid technique was used to explore complementary studies courses when compared to other engineering courses within the same curriculum. The relationships between elements and constructs in the grids were analysed using the repertory grid techniques of principal component analysis and cluster analysis. What became clear was that while most of the educators interviewed did recognise complementary studies courses as different to courses considered as core or technical, what made them different was very unclear. Each educator had a very different conception of what defines, differentiates or constitutes a complementary studies course. This range of variation may go some way to explaining why complementary courses seem out of place in engineering programs by educators and students alike