Integrating technology into mathematics teaching at the university level

The emergence of new computing technologies in the second half of the twentieth century brought about new potentials and promised the rapid transformation of the teaching and learning of mathematics. However, despite the vast investments in technology resources for schools and universities, the realities of schooling and the complexities of technology-equipped environments resulted in a much slower integration process than was predicted in the 1980s. Hence researchers, together with teachers and mathematicians, began examining and reflecting on various aspects of technology-assisted teaching and learning and on the causes of slow technology integration. Studies highlighted that as technology becomes increasingly available in schools, teachers’ beliefs and conceptions about technology use in teaching are key factors for understanding the slowness of technology integration. In this paper, I outline the shift of research focus from learning and technology environment-related issues to teachers’ beliefs and conceptions. In addition, I highlight that over the past two decades a considerable imbalance has developed in favour of school-level research against university-level research. However, several changes in universities, such as students declining mathematical preparedness and demands from other sciences and employers, necessitate closer attention to university-level research. Thus, I outline some results of my study that aimed to reflect on the paucity of research and examined the current extend of technology use, particularly Computer Algebra Systems (CAS) at universities, mathematicians’ views about the role of CAS in tertiary mathematics teaching, and the factors influencing technology integration. I argue that due to mathematicians’ extensive use of CAS in their research and teaching, documenting their teaching practices and carrying out research at this level would not only be beneficial at the university level but also contribute to our understanding of technology integration at all levels.     

Towards an authentic teaching of mathematics: Hans-Georg Steiner’s contribution to the reform of mathematics teaching

Hans-Georg Steiner was the “motor of the reform” of mathematics education in Germany. His main concern was to promote authentic teaching. His suggestions for teaching mathematical structures stimulated the process of reform, but were criticised as well. Two controversies are studied in this paper. The controversy with Detlef Laugwitz in 1965 was about the dichotomy “axiomatics vs. constructiveness”. Another controversy with Alexander Wittenberg in 1964 was about the problem of “elementary”. The following considerations can show the need for fundamental didactical analyses in mathematics education, as they were initiated by Hans-Georg Steiner.     

Improving resource utilization for diagnostic services through flexible inpatient scheduling: A method for improving resource utilization

We present a method to increase the utilization of and reduce the waiting times for an under-capacitated diagnostic resource in the presence of uncertain demand with several priority levels. We consider the case of a computed tomography (CT) scanning department that services both high-priority in-patients and lower priority outpatients. Current practice calls for all in-patient demand to be met on the day of the request. Our proposal looks at the benefit of reserving space for carrying over a percentage of non-emergency in-patient demand to the next day and utilizing a pool of on-call outpatients who can respond quickly to available capacity. We formulate and solve an optimization problem that returns a reservation policy that minimizes unused capacity subject to an overtime constraint. We use a simulation to demonstrate a significant reduction in the growth rate of outpatient waiting time resulting from using the proposed method and investigate the sensitivity of results to several model assumptions.     

On the student evaluation of university courses and faculty members’ teaching performance

Trying to determine higher education quality, one gets quickly to one of its significant dimensions, namely the quality of faculty members’ teaching. The latter and, overall, the quality of any university course should be certainly evaluated by their recipients, namely students. In this paper we develop a statistical framework based on Statistical Quality Control mainly, which can be used in order to exploit student evaluations as much as possible. More specifically we present two directions of data monitoring and analysis; the one uses control charts and the other hypotheses testing. The results that can be raised through both directions are crucial for any decision maker.     

Maximum matchings in random bipartite graphs and the space utilization of Cuckoo Hash tables

We study the the following question in Random Graphs. We are given two disjoint sets L,R with |L| = n and |R| = m. We construct a random graph G by allowing each x∈L to choose d random neighbours in R. The question discussed is as to the size μ(G) of the largest matching in G. When considered in the context of Cuckoo Hashing, one key question is as to when is μ(G) = n whp? We answer this question exactly when d is at least three. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Policy issues in the teaching and learning of the mathem atical sciences at university level

Policy decisions and their implementation impact on the teaching and learning of mathematics in many ways. Sometimes this is overt and takes account of local factors and professional opinion. However global economic policies, often dominated by the actions of huge multinational corporations, can have considerable influence on all aspects of education including mathematics education. These policies appear to be creating greater inequity between and within nations. Quality mathematics education for all, a recurring theme at International Congress on Mathematical Education meetings, is becoming less of a reality. At the same time, mathematics education at every level is increasingly influenced by powerful bureaucrats rather than by the profession. It is suggested that mathematical scientists need to try to understand the political forces affecting mathematics education for the ICMI study on teaching and learning mathematics at the tertiary level to have maximum impact.     

Beyond statistical methods: teaching critical thinking to first-year university students

We discuss a major change in the way we teach our first-year statistics course. We have redesigned this course with emphasis on teaching critical thinking. We recognized that most of the students take the course for general knowledge and support of other majors, and very few are planning to major in statistics. We identified the essential aspects of a first-year statistics course, given this student mix, focusing on a simple question, ‘Given this is the last chance you have to teach statistics, what are the essential skills students need?’ We have moved from thinking about statistics skills needed for a statistician to skills needed to participate in today's society. We have changed the way we deliver the course with less emphasis on lectures and more on alternative resources including on-line tutorials, Excel, computer-based skills testing, web-based learning materials and smaller group activities such as study groups and example classes. Feedback from students shows that they are very receptive and enthusiastic.     

The analysis of the understanding of the three-dimensional (Euclidian) space and the two-variable function concept by university students

Conceptual understanding is being emphasized in mathematics education. Students often have difficulty understanding the multi-variable function, a key concept. Based on the APOS theory, which analyzes the cognitive structures formed by individuals in learning a mathematical concept and produces components related to that learning, this study analyzes the conceptual understanding of three-dimensional spaces and two-variable functions by university students. The genetic decomposition of these concepts proposed by Trigueros and Martinez-Planell is also considered. The analyzes results revealed that only one student constructed the concept of three-dimensional space as an object within the framework of genetic decomposition. Some students could not relate the concepts of two-variable function and three-dimensional space. Students who could perform algebraic operations had problems related to geometric representation. This study suggests the refinement of genetic decomposition to include, e.g., mental construction steps for writing algebraic equations of special surfaces whose graphs are given in R³.     

The history of mathematics as a coupling link between secondary and university teaching

During the years they spend in university, many mathematics students develop a very poor conception of mathematics and its teaching. This fact is bad in all cases, but even more in the case of those students who will be mathematics teachers in school. In this paper it is argued that the history of mathematics may be an efficient element to provide students with flexibility, open-mindedness and motivation towards mathematics. The theoretical background of this work relies both on recent research in mathematics education and on papers written by mathematicians of the past. Opinions are supported with examples. One example concerns a historical presentation of ‘definition’; it was developed with mathematics students who will become mathematics teachers. For students oriented to research or to applied mathematics, an example is presented to address the problem of the secondary-tertiary transition.     

Mathematical working space and paradigms as an analysis tool for the teaching and learning of analysis

ZDM – Mathematics Education - Mathematical working space (MWS) is a model that is used in research in mathematics education, particularly in the field of geometry. Some MWS elements are...     

On a research programme concerning the teaching and learning of linear algebra in the first-year of a French science university

The paper discusses the current approach to the teaching of linear algebra in the first year at a French science university and the main difficulties that students have with this material. A brief account is given of the first steps towards the design of a teaching experiment. From a joint didactical and historical survey a first hypothesis is drawn: epistemological specificity, the use of ‘meta-lever’, the use of changes of settings and points of view, and the importance of the concept of rank. The main aspects and objectives of the teaching design with which we experimented over a whole teaching semester for five years with around 200 students are presented. Finally, the type of evaluations that were set up and the difficulties encountered are explained. The conclusion deals with issues on the teaching and learning of linear algebra as well as issues on methodological and theoretical points in relation to the original didactical framework.     

On the Schwartz space of the basic affine space

Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G / U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct a certain canonical G-invariant space (called the Schwartz space of X) of functions on X, which is an extension of the space of smooth compactly supported functions on X. We show that the space of all elements of , which are invariant under the Iwahori subgroup I of G, coincides with the space generated by the elements of the so called periodic Lusztig basis, introduced recently by G. Lusztig (cf. [10] and [11]). We also give an interpretation of this space in terms of a certain equivariant K-group (this was also done by G. Lusztig — cf. [12]). Finally we present a global analogue of , which allows us to give a somewhat non-traditional treatment of the theory of the principal Eisenstein series.     

Wasserstein space over the Wiener space

The goal of this paper is to study optimal transportation problems and gradient flows of probability measures on the Wiener space, based on and extending fundamental results of Feyel–Üstünel. Carrying out the program of Ambrosio–Gigli–Savaré, we present a complete characterization of the derivative processes for certain class of absolutely continuous curves. We prove existence of the gradient flow curves for the relative entropy w.r.t. the Wiener measure and identify these gradient flow curves with solutions of the Ornstein–Uhlenbeck evolution equation.     

A container yard storage strategy for improving land utilization and operation efficiency in a transshipment hub port

This paper studies the storage yard management problem in a busy transshipment hub, where intense loading and unloading activities have to be considered at the same time. The need to handle huge volumes of container traffic and the scarcity of land in the container port area pose serious challenges for the port operator to provide efficient services. A consignment strategy with a static yard template has been used to reduce the level of reshuffles in the yard, but it sacrifices on land utilization because of exclusive storage space reservation. Two space-sharing approaches are proposed to improve on the land utilization through dynamic reservation of storage space for different vessels during different shifts. Meanwhile, workload assignment among reserved spaces will also satisfy the high-low workload balancing protocol to reduce traffic congestion in the yard. A framework which integrates space reservation and workload assignment is proposed. Experimental results show that the framework is able to provide solutions for containers handling within much less storage space, while guarantee the least yard crane deployment.     

A multidimensional approach to training mathematics students at a university: improving the efficiency through the unity of social,psychological and pedagogical aspects

The article deals with social, psychological and pedagogical aspects of teaching mathematics students at universities. The sociological portrait and the factors influencing a career choice of a mathematician have been investigated through the survey results of 198 first-year students of applied mathematics major at 27 state universities (Russia). Then, psychological characteristics of mathematics students have been examined based on scientific publications. The obtained results have allowed us to reveal pedagogical conditions and specific ways of training mathematics students in the process of their education at university. The article also contains the analysis of approaches to the development of mathematics education both in Russia and in other countries. The results may be useful for teaching students whose training requires in-depth knowledge of mathematics.     

Towards the reconstruction of posets

The reconstruction conjecture for posets is the following: Every finite posetP of more than three elements is uniquely determined — up to isomorphism — by its collection of (unlabelled) one-element-deleted subposets P–{x}:xV(P).We show that disconnected posets, posets with a least (respectively, greatest) element, series decomposable posets, series-parallel posets and interval orders are reconstructible and that N-free orders are recognizable.We show that the following parameters are reconstructible: the number of minimal (respectively, maximal) elements, the level-structure, the ideal-size sequence of the maximal elements, the ideal-size (respectively, filter-size) sequence of any fixed level of the HASSE-diagram and the number of edges of the HASSE-diagram.This is considered to be a first step towards a proof of the reconstruction conjecture for posets.Research partly supported by DAAD.     

Towards the Willmore conjecture

We develop a variety of approaches, mainly using integral geometry, to proving that the integral of the square of the mean curvature of a torus immersed in must always take a value no less than . Our partial results, phrased mainly within the -formulation of the problem, are typically strongest when the Gauss curvature can be controlled in terms of extrinsic curvatures or when the torus enjoys further properties related to its distribution within the ambient space (see Sect. 3). Corollaries include a recent result of Ros [20] confirming the Willmore conjecture for surfaces invariant under the antipodal map, and a strengthening of the expected results for flat tori. The value arises in this work in a number of different ways – as the volume (or renormalised volume) of or , and in terms of the length of shortest nontrivial loops in subgroups of SO(4). Received April 26, 1999 / Accepted January 14, 2000 / Published online June 28, 2000     

Erratum to: Mathematical working space and paradigms as an analysis tool for the teaching and learning of analysis

ZDM – Mathematics Education -     

On the embedding of BV space into Besov-Orlicz space

We give a sufficient (and, in the case of a compact domain, a necessary) condition for the embedding of Sobolev space of functions with integrable gradient into Besov-Orlicz spaces to be bounded. The condition has a form of a simple integral inequality involving Young and weight functions. We provide an example with Matuszewska-Orlicz indices of involved Orlicz norm equal to one. The main tool is the molecular decomposition of functions from a BV space.