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.     

The computer-related sciences (synnoetics) at a university in the year 1975

BIT Numerical Mathematics -     

A metacognitive intervention in mathematics at university level

The paper reports the main results of an instructional study. The study was aimed at improving the performance in mathematics of a group of university students of biology who repeatedly failed the final examination of a compulsory course in mathematics. The main difficulties of these students seemed to be metacognitive and affective in nature. The training therefore worked on metacognitive and affective features: knowledge about cognition, monitoring, beliefs, emotions and attitudes. The intervention was successful: at the end of the course all students passed the examination that they had failed so often. The results also suggest that it may be possible (and necessary) to ‘teach learning to learn’ mathematics.     

Connected functional working spaces: a framework for the teaching and learning of functions at upper secondary level

ZDM – Mathematics Education - This paper aims at contributing to remedy the narrow treatment of functions at upper secondary level. Assuming that students make sense of functions by working...     

The student experience of mathematical proof at university level

While proofs are central to university level mathematics courses, research indicates that some students may complete their degrees with an incomplete picture of what constitutes a proof and how proofs are developed. The paper sets out to review what is known of the student experience of mathematical proof at university level. In particular, some evidence is presented of the conceptions of mathematical proof that recent mathematics graduates bring to their postgraduate course to teach high school mathematics. Such evidence suggests that while the least well-qualified graduates may have the poorest grasp of mathematical proof, the most highly qualified may not necessarily have the richest form of subject matter knowledge needed for the most effective teaching. Some indication of the likely causes of this incomplete student perspective on proof are presented.     

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.     

Redesigning the calculus sequence at a research university: issues,implementation, and objectives

The paper discusses the progress and challenges of a new reformed calculus sequence for science, engineering, and mathematics students developed by the Institute of Technology Centre for Educational Programs and School of Mathematics, University of Minnesota. The main objective of the Initiative is to enable undergraduates to better learn calculus and the critical thinking skills necessary to apply it in a variety of science and engineering problems. Changes in content and pedagogy are emphasized, including instructional teamwork and student-centred learning, involving students working cooperatively in small groups and exploring mathematical ideas using appropriate technologies. Achievement and retention of Initiative students are compared with a control group from the standard calculus sequence. Student attitudes about the usefulness of the Initiative's curriculum, pedagogy, and its influence on learning are discussed. Future implications including new uses of distributed learning are also addressed.     

Assessment of learning in university mathematics

The important issues in assessment practices for university undergraduates are identified. The way in which assessment can be used to enhance student learning, the impact of external factors on assessment methods and the barriers that inhibit change are discussed. The paper also discusses the various ways in which changes in assessment practices have have been implemented and studies that have been carried out to gauge the effect of different methods of assessment.     

Integrating history and philosophy in mathematics education at university level through problem-oriented project work

Through the last three decades several hundred problem-oriented student-directed projects concerning meta-aspects of mathematics and science have been performed in the 2-year interdisciplinary introductory science programme at Roskilde University. Three selected reports from this cohort of project reports are used to investigate and present empirical evidence for learning potentials of integrating history and philosophy in mathematics education. The three projects are: (1) a history project about the use of mathematics in biology that exhibits different epistemic cultures in mathematics and biology. (2) An educational project about the difficulties of learning mathematics that connects to the philosophy of mathematics. (3) A history of mathematics project that connects to the sociology of multiple discoveries. It is analyzed and discussed in what sense students gain first hand experiences with and learn about meta-aspects of mathematics and their mathematical foundation through the problem-oriented student-directed project work.     

Qualities of examples in learning and teaching

In this paper, we theorise about the different kinds of relationship between examples and the classes of mathematical objects that they exemplify as they arise in mathematical activity and teaching. We ground our theorising in direct experience of creating a polynomial that fits certain constraints to develop our understanding of engagement with examples. We then relate insights about exemplification arising from this experience to a sequence of lessons. Through these cases, we indicate the variety of fluent uses of examples made by mathematicians and experienced teachers. Following Thompson’s concept of “didactic object” (Symbolizing, modeling, and tool use in mathematics education. Kluwer, Dordrecht, The Netherlands, pp 191–212, 2002), we talk about “didacticising” an example and observe that the nature of students’ engagement is important, as well as the teacher’s intentions and actions (Thompson avoids using a verb with the root “didact”. We use the verb “didacticise” but without implying any connection to particular theoretical approaches which use the same verb.). The qualities of examples depend as much on human agency, such as pedagogical intent or mathematical curiosity or what is noticed, as on their mathematical relation to generalities.     

The teaching of proof at the lower secondary level—a video study

Teaching mathematical proof is one of the most challenging topics for teachers. Several empirical studies revealed repeatedly different kinds of students’ problems in this area. The results give support that students’ abilities in proving are significantly influenced by their specific mathematics classrooms. In this paper we will present a method for evaluating proof instruction and some results of a video study that describe proving processes in mathematics classrooms at the lower secondary level from a mathematical perspective.     

Towards improving the utilization of university teaching space

There is a perception that teaching space in universities is a rather scarce resource. However, some studies have revealed that in many institutions it is actually chronically under-used. Often, rooms are occupied only half the time, and even when in use they are often only half full. This is usually measured by the ‘utilization’ which is defined as the percentage of available ‘seat-hours’ that are employed. Within real institutions, studies have shown that this utilization can often take values as low as 20–40%. One consequence of such a low level of utilization is that space managers are under pressure to make more efficient use of the available teaching space. However, better management is hampered because there does not appear to be a good understanding within space management (near-term planning) of why this happens. This is accompanied, within space planning (long-term planning) by a lack of experise on how best to accommodate the expected low utilizations. This motivates our two main goals: (i) To understand the factors that drive down utilizations, (ii) To set up methods to provide better space planning. Here, we provide quantitative evidence that constraints arising from timetabling and location requirements easily have the potential to explain the low utilizations seen in reality. Furthermore, on considering the decision question ‘Can this given set of courses all be allocated in the available teaching space?’ we find that the answer depends on the associated utilization in a way that exhibits threshold behaviour: There is a sharp division between regions in which the answer is ‘almost always yes’ and those of ‘almost always no’. Through analysis and understanding of the space of potential solutions, our work suggests that better use of space within universities will come about through an understanding of the effects of timetabling constraints and when it is statistically likely that it will be possible for a set of courses to be allocated to a particular space. The results presented here provide a firm foundation for university managers to take decisions on how space should be managed and planned for more effectively. Our multi-criteria approach and new methodology together provide new insight into the interaction between the course timetabling problem and the crucial issue of space planning.     

The role of prediction in the teaching and learning of mathematics

The prevalence of prediction in grade-level expectations in mathematics curriculum standards signifies the importance of the role prediction plays in the teaching and learning of mathematics. In this article, we discuss benefits of using prediction in mathematics classrooms: (1) students’ prediction can reveal their conceptions, (2) prediction plays an important role in reasoning and (3) prediction fosters mathematical learning. To support research on prediction in the context of mathematics education, we present three perspectives on prediction: (1) prediction as a mental act highlights the cognitive aspect and the conceptual basis of one's prediction, (2) prediction as a mathematical activity highlights the spectrum of prediction tasks that are common in mathematics curricula and (3) prediction as a socio-epistemological practice highlights the construction of mathematical knowledge in classrooms. Each perspective supports the claim that prediction when used effectively can foster mathematical learning. Considerations for supporting the use of prediction in mathematics classrooms are offered.     

Appropriate use of a CAS in the teaching and learning of mathematics

If the use of a computer algebra system (CAS) is to be meaningful and have an impact on students, then it must be grounded in good pedagogy and have some clearly defined goals. It is the authors' belief that an important goal for teaching mathematics with the CAS is that courses be designed so that students can become active participants in their learning experience, planning the problem-solving strategies and carrying them out. The CAS becomes an important tool and a partner in this learning process. To this end, here the authors' have linked the use of the CAS to an existing classification scheme for Mathematical Tasks, called the MATH Taxonomy, and illustrated, through concrete examples, how the goals of teaching and learning of mathematics can be set using this classification together with the CAS.     

Challenges and opportunities for the implementation of inquiry-based learning in day-to-day teaching

In this paper, we analyse the conditions and constraints which might favour, or on the contrary hinder, a large-scale implementation of inquiry-based mathematics and science education, on the basis of our work within the PRIMAS project in 12 European countries. As a complement to the approach through the analysis of teachers’ beliefs and practices (see Engeln et al. in ZDM Int J Math Educ 45(6), this issue, 2013), we tackle this issue from a systemic institutional perspective. Indeed, in our approach, we consider teachers as actors of institutions, representing some disciplines, embedded in a school system, sharing some common pedagogical issues, in relation to society. Our sources of information are easily accessible public documents. With a theoretical background from Chevallard’s anthropological theory of didactics, we organized our analysis according to four levels of institutional organization that co-determine both content and didactical aspects in the teaching of mathematics and sciences: society, school, pedagogy and disciplinary. Our approach is systemic in the sense that we do not focus on teachers as individuals, nor on the curricula, the organization of teachers’ training or the textbooks themselves. Rather, we trace the way the conditions and constraints are operative, provide the main results of our analysis and draw out a few perspectives according to our four levels of didactical determination. Finally, in the conclusion, we reflect on the limits and potential of our analysis.