Implementing a flipped classroom approach in a university numerical methods mathematics course

This paper describes and analyses the implementation of a ‘flipped classroom’ approach, in an undergraduate mathematics course on numerical methods. The approach replaced all the lecture contents by instructor-made videos and was implemented in the consecutive years 2014 and 2015. The sequential case study presented here begins with an examination of the attitudes of the 2014 cohort to the approach in general as well as analysing their use of the videos. Based on these responses, the instructor makes a number of changes (for example, the use of ‘cloze’ summary notes and the introduction of an extra, optional tutorial class) before repeating the ‘flipped classroom’ approach the following year. The attitudes to the approach and the video usage of the 2015 cohort are then compared with the 2014 cohort and further changes that could be implemented for the next cohort are suggested.     

Teaching mathematical modelling through project work

The paper presents and analyses experiences from developing and running an inservice course in project work and mathematical modelling for mathematics teachers in the Danish gymnasium, e.g. upper secondary level, grade 10~12. The course objective is to support the teachers to develop, try out in their own classes, evaluate and report a project based problem oriented course in mathematical modelling. The in-service course runs over one semester and includes three seminars of 3, 1 and 2 days. Experiences show that the course objectives in general are fulfilled and that the course projects are reported in manners suitable for internet publication for colleagues. The reports and the related discussions reveal interesting dilemmas concerning the teaching of mathematical modelling and how to cope with these through «setting the scene» for the students modelling projects and through dialogues supporting and challenging the students during their work. This is illustrated and analysed on the basis of two course projects.     

数学教学中思维的拓展与延伸

结合五个教学案例说明教师在教学中如何依据教学内容进行思维的拓展与延伸,从而改善学生的思维品质与思维习惯.     

Generalized nets in neurology: an example of mathematical modelling

The purpose of this paper is to explain the basic theory of generalized nets (GNs) and their applications in the context of the differential diagnosis of neurological diseases. We define formally the concepts of a GN and transitions of a GN and also outline some remarks on their theory. The work here constructs an example which aims to trace the process of diagnosing different signs and symptoms in neurology. This will enable the interested reader to see the scope of nets in general as tools for the modelling, simulation, optimization and control of real processes.     

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.     

Technology-active mathematical modelling

Problems in mathematical modelling and data analysis are discussed from a constructivist perspective. This approach provides students with realistic opportunities to connect mathematics to significant social and environmental problems while incorporating recent advances made possible by today's mathematically powerful calculators. Also included are methods for enhancing students' abilities to shift among a wide range of representations using the modelling capabilities in graphing utilities. Consideration is further given to the changes that technology imposes on the classroom culture, including changes in students' attitudes about modelling techniques and difficulties in locating appropriate problems. The article concludes by discussing the integration of teaching and assessment with mathematical modelling.     

A role for M -matrices in modelling population growth

Adopting a discrete-time cohort-type model to represent the dynamics of a population, the problem of achieving a desired total size of the population under a balanced growth (contraction) and the problem of maintaining the desired size, once achieved, are studied. Properties of positive-time systems and M-matrices are used to develop the results, which are illustrated using simple examples. The material is presented in a format that makes it appropriate as background material for interesting applications based modelling assignments on undergraduate programmes in the mathematical sciences, applied sciences, economics and technology. All relevant properties of both positive-time systems and M-matrices are presented and clarified at the outset.     

数学实验课程的教学与数学素质的培养

分析了现行工科数学教学中存在的主要问题 ,认为数学实验课程的教学是培养数学素质不可缺少的环节 ,并就如何更好地发挥数学实验课程教学的作用提出了一些具体的建议 .     

Dynamic investigation of loci with surprising outcomes and their mathematical explanations

The locus is a very important concept in Euclidean geometry since it serves as a tool for solving different problems, and allows geometric constructions to be carried out. The teaching of the subject of loci in various mathematics courses includes solution of different exercises in which the student is required to find the locus in accordance with the data of the question. The present paper offers a different view of the subject of loci, which brings about conceptual understanding of the subject with identification of conserved properties and suitable generalizations obtained through investigation that includes the use of dynamic geometric software (GeoGebra). General formulas were developed for the equation of the locus in two cases. In the article, there are links to geometric applets which allow one to demonstrate the loci formed in some cases.     

Understanding authority in small-group co-constructions of mathematical proof

Authority becomes shared in mathematics classrooms when perceived sources of valid mathematical knowledge extend beyond the teacher/textbook and allow both students and disciplinary modes of reasoning to hold authority. The goal of this research is to better understand classroom situations that are intended to facilitate shared authority over proof, namely small-group episodes where students are granted authority (Gerson & Bateman, 2010) to co-construct mathematical proofs. We sought to better understand the content of undergraduate students’ attention during group proving and the sources of legitimacy for students. Using Stylianides’ (2007) definition of proof as an analytical frame, we found that student discourse focused primarily upon the mode of argumentation, followed by the mode of argument representation, and then the set of accepted statements. We identified four themes with respect to the sources of authority students relied upon in their group proving: (1) the course rubric, (2) peers’ confidence, (3) form and symbols, and (4) logical structure. Implications for research and practice are presented.     

Pedagogical challenges in making mathematics relevant for Indigenous Australians

After the invasion and occupation of their land by Europeans, Indigenous Australians were expected to benefit from a Western education system based on that of the dominant Europeans. Gradually educators realized that Aboriginal children learn differently and that Indigenous culture and pedagogy has validity and strength. Of course, educators also need to be acutely aware of the diversity of Indigenous cultures in Australia and that there is not a monolithic sense of Aboriginal identity or Aboriginal pedagogy. Dispossession and alienation, poor health and few employment opportunities must also affect educational interest, attendance, application and performance. In this paper the characteristics of Indigenous learners are examined and pedagogical strategies to assist in both students' learning and teachers' delivery are explored. The message conveyed in this paper has particular relevance for teaching mathematics to Aboriginal learners. It is also of considerable value in teaching mathematics to nonAboriginal students, in Australia and elsewhere.     

An investigation into the use of graphics calculators with pupils in Key Stage 2

This paper outlines an experiment in which pupils in Key Stage 2 were encouraged to use graphics calculators, in particular two simple programs, which helped them develop recall of their tables and allowed them to practise multiplication. The pupils responded very well to the calculators and seemed to have been motivated by them. The pupils did not find them difficult to operate and experienced very few technical problems. The authors concluded that the graphics calculator has considerable potential to enhance the mathematical experience and learning of pupils at this level, and that although the extent of the investigation was fairly limited the results were encouraging enough to justify further work in this area.     

Applied mathematical modelling in marine science

The essential elements of a general interdisciplinary marine model are reviewed; state variables are identified, evolution equations are discussed and major parameters are indicated. The methods of reducing the size of the model by aggregation in state space (reduction of scope) or averaging in physical space (reduction of support) are revised with particular emphasis on the associated parameterization of the non-linear effects. Examples of application are given in illustration and comparison of the models' predictions with observations indicates the degree of accuracy and the limits of viability of present state marine modelling.     

An investigation of assessment and feedback practices in fully asynchronous online undergraduate mathematics courses

Research suggests it is difficult to learn mathematics in the fully asynchronous online (FAO) instructional modality, yet little is known about associated teaching and assessment practices. In this study, we investigate FAO mathematics assessment and feedback practices in particular consideration of both claims and findings that these practices have a powerful influence on learning.

A survey questionnaire was constructed and completed by 70 FAO undergraduate mathematics instructors, mostly from the USA, who were each asked to detail their assessment and feedback practices in a single FAO mathematics course. Alongside these questions, participants also answered the 16-item version of the Approaches to Teaching Inventory. In addition, a novel feedback framework was also created and used to examine how feedback practices may be related to participants' approaches to teaching.

Results show that assessment and feedback practices are varied and complex: in particular, we found there was not a simple emphasis on summative assessment instruments, nor a concomitant expectation these would always be invigilated. Though richer assessment feedback appears to be emphasized, evidence suggests this feedback may not be primarily directed at advancing student learning. Moreover, we found evidence of a reliance on computer--human interactions (e.g. via computer-assisted assessment systems) and further evidence of a decline in human interactions, suggesting a dynamic that is both consistent with current online learning theory and claims FAO mathematics courses are becoming commodified. Several avenues for further research are suggested.     

An efficacious dynamic mathematical modelling approach for creation of best collaborative groups

In many disciplines, including business, publishing, management, health, sports, arts and education, there is a population of people which should be optimally divided into multiple groups based on certain attributes to collaboratively perform a particular task. The problem becomes more complex when some other requirements are also added. They might be importance degrees of grouping criteria, homogeneity, heterogeneity or a mixture of teams, amount of consideration to the preferences of individuals, variability or invariability of group size, having moderators, aggregation or distribution of persons, overlapping level of teams, and so forth. Several researchers have addressed the problem, but they suffered from failure to satisfy all the requirements and/or developed inexact solutions and/or had very long process times. This work reveals how these problems can be mathematically formulated through a binary integer programming approach to construct an effective model which is solvable by exact methods in an acceptable time. The suggested model was validated through data obtained from collaboration of a set of learners in an online learning discussion forum grouped by means of the provided method. The achieved outcomes confirmed that the new approach is satisfactory and promising.     

Rival approaches to mathematical modelling in immunology

In order to formulate quantitatively correct mathematical models of the immune system, one requires an understanding of immune processes and familiarity with a range of mathematical techniques. Selection of an appropriate model requires a number of decisions to be made, including a choice of the modelling objectives, strategies and techniques and the types of model considered as candidate models. The authors adopt a multidisciplinary perspective.     

Teacher flexibility in mathematical discussion

The significance of discussion in mathematics classes has been prominently debated in the research literature. Different studies have stressed the importance of teacher flexibility in orchestrating the discussion. We introduce an operational definition of teacher flexibility. In a case study with one secondary-school mathematics teacher, we microanalyzed discussion situations in which the teacher had to change her plan according to unforeseen student replies. The analysis was aimed at characterizing situations in which the teacher was either flexible or inflexible in her interactions with students and describing the factors that affected her flexibility. We suggest four basic patterns of teacher flexibility and discuss the complexity of the factors that shape them.     

高等数学教学现状调查分析

高等学校扩大招生后,非数学专业的数学教育面临很多问题,大部分人把这些问题归结为学生质量的下降造成的,这是不公正的.教学过程是一个系统的过程,从对1995级和2005级同一份高等数学(下册)试卷的考试情况进行分析,虽然可以显现出学生的差异,但这些差异不全是学生的质量的问题,应该与教学环境、教学要求、教学方法、师资条件以及社会发展环境等相关联,同时与学生的学习兴趣、学习能力是密不可分的.调查分析这些因素对当代学生的学习和数学教学的影响程度到底有多大,是有效提高大学数学教学质量的前提.