Teaching Mathematics and its Applications - recent issues

A note on arc length

Patyi, I. — Mon, 24 Aug 2009 22:49:01 PDT

We consider how the arc length integral of the graph of a function in the plane is connected with the hyperbola and its rational parameterization.

A proof of the converse of the Pythagorean proposition

Scimone, A. — Mon, 24 Aug 2009 22:49:01 PDT

The article presents a demonstration of the converse of the Pythagorean Theorem based on the reductio ad absurdum. This is necessary to overcome the discrepancy, noticed by pupils, between the Euclidean purpose to demonstrate that the given triangle is right-angled and the auxiliary figure by which the given triangle is drawn as if it were already a right-angled one. To the eyes of students this does not make the Euclidean reasoning clear.

The impact of the mathematics support centre on the grades of first year students at the National University of Ireland Maynooth

Bhaird, C. M. a., Morgan, T., O'Shea, A. — Mon, 24 Aug 2009 22:49:01 PDT

In this article, we consider the mathematics grades of first year students at the National University of Ireland Maynooth and the influence that the Mathematics Support Centre (MSC) has on these grades. We will consider the evidence to suggest that the MSC has a positive effect on the grades of the students who attend the centre. It seems to be particularly beneficial to students with weak mathematical backgrounds. As these students are most at risk of failing or dropping out of University, the positive impact of the MSC on their grades is very encouraging.

An application of calculus: optimum parabolic path problem

Atasever, M., Pakdemirli, M., Yurtsever, H. A. — Mon, 24 Aug 2009 22:49:01 PDT

A practical and technological application of calculus problem is posed to motivate freshman students or juniory high school students. A variable coefficient of friction is used in modelling air friction. The case in which the coefficient of friction is a decreasing function of altitude is considered. The optimum parabolic path for a flying object for which the work done by variable coefficient air friction force is minimum is determined. If a flying object follows the determined path, energy requirement would be less compared with other parabolic or linear paths.

An integrative learning experience within a mathematics curriculum

Melendez, B., Bowman, S., Erickson, K., Swim, E. — Mon, 24 Aug 2009 22:49:02 PDT

We developed four separate scenarios focusing on the connections between mathematics, biology, and social sciences. This structure facilitated the coordination of faculty from seven academic departments on campus. Each scenario had students from different majors build mathematical models, gather information from their respective disciplines, and develop a final presentation that included a committee consensus on how to approach the problem in a practical way. As a result, students learned how mathematics plays a role in other disciplines and how insight from different points of view affects the approach taken to a complex problem.

Teaching mathematics understandings for transfer

Jones, J. L., Jones, K. A., Vermette, P. J. — Mon, 24 Aug 2009 22:49:02 PDT

Promoting student understanding for transfer is an illusive hallmark of effective mathematics instruction. While much research has shown the necessity of promoting understanding for transfer, less attention has been paid to actual pedagogical strategies that can be used to promote transfer of mathematical ideas. Using Fogarty et al. (1992, How to Teach for Transfer, The mindful school, Palatine, Illinois: Skylight Publishing) ‘ten tools for teaching for transfer’ as a model, the authors of this piece provide numerous suggestions designed specifically for promoting transfer of concepts in secondary mathematics.

The immortal ant and the expanding balloon

Griffiths, M. — Mon, 24 Aug 2009 22:49:02 PDT

In this article we consider, via a specific modelling example, the educational benefits to be gained from running mathematical activities with our sixth-form and undergraduate students that, in modern parlance, might be termed ‘rich tasks’. The idea for this modelling activity arose while the author was reading a popular-science book on cosmology (in particular, on the possible shapes of the universe). Light travelling around the universe was likened to an ant crawling around a balloon. A statement in the book regarding the ant's progress around the balloon did not entirely ring true with the author, and his subsequent investigations led to the activity described here. We explore several scenarios associated with the model in order both to pre-empt possible paths taken by the students and to be able to provide some guidance when necessary. Suggestions are given as to how the activity may be extended, and then, after highlighting the numerous educational benefits, we consider the potential pitfalls and difficulties associated with the delivery of tasks such as these.

Real world graph connectivity

Lind, J., Narayan, D. — Mon, 24 Aug 2009 22:49:02 PDT

We present the topic of graph connectivity along with a famous theorem of Menger in the real-world setting of the national computer network infrastructure of National LambdaRail. We include a set of exercises where students reinforce their understanding of graph connectivity by analysing the National LambdaRail network. Finally, we give suggestions for using this module as a project in an undergraduate mathematics course.

Does students' confidence in their ability in mathematics matter?

Parsons, S., Croft, T., Harrison, M. — Mon, 25 May 2009 06:46:14 PDT

Research was conducted into first year engineering students’ learning of mathematics in a university college during 2005–2007. The aims were to understand better students’ confidences and explore which factors affected performance and how these were inter-related. Questionnaires were administered which posed questions regarding previous mathematics qualifications, student confidences, attitude, liking of the subject and motivation. The responses were analysed and compared with marks achieved by the students in their first year engineering mathematics examinations. The majority of students were fairly confident, reported improved confidence acquired during their first year of university study and had positive attitudes. Better mathematically qualified students were generally more confident and successful in mathematics. A regression model was produced which predicted a 12% increase in mathematics marks per increase in GCSE mathematics grade, and 5% increase in marks for each increase in confidence level. Thus, better qualifications (and the skills represented) were shown to be associated with better university marks and student confidence also produced a notable association with the marks achieved. The findings suggest that having attended to the mathematics syllabi, lecturers could seek to boost student confidence in their ability in mathematics as a further means to improve student performance at university.

GeoGebra -- freedom to explore and learn

Fahlberg-Stojanovska, L., Stojanovski, V. — Mon, 25 May 2009 06:46:14 PDT

We start by visiting the maths section of the web site answers.yahoo.com. Here, anybody can ask a question from anywhere in the world at every possible level. Answers are given by anyone who wants to contribute and then askers/readers rate the responses. A brief look here and it is starkly clear that our young people are struggling and their ability to think logically—that is understand a problem, organize data into knowns and unknowns, explore possibilities and assess solutions is definitely on the decline. In our opinion, this is more insidious than the actual decline in their overall mathematics skills. Further, one is struck by the fact that technology seems to be contributing to this decline when in fact it should be the opposite. We then examine two question/answer cycles in detail and show how the freeware GeoGebra (www.geogebra.org GeoGebraWiki: www.geogebra.org/wiki GeoGebraForum: www.geogebra.org/forum)—which gives the freedom to explore and learn to everyone, everywhere and at any time—can be of tremendous value to pupils and students in their understanding of mathematics from the smallest ages on up.

Factors influencing the transition to university service mathematics: part 1 a quantitative study

Liston, M., O'Donoghue, J. — Mon, 25 May 2009 06:46:14 PDT

This article reports on a quantitative study carried out into the influence of affective variables, the role of conceptions of mathematics and approaches to learning on students in the transition to Service mathematics at the University of Limerick (UL). Questionnaires were distributed to three groups of first year students on Service mathematics programmes (degree courses where mathematics plays a part in the students’ studies but is not the main focus) at UL, at the beginning of the university academic year 2006/2007. Their attitudes, beliefs, self-concept, conceptions of mathematics and approaches to learning are examined and relationships between specific variables are reported on. The impact these concepts, as well as final secondary school mathematics exam results, have on performance are also discussed.

Change in senior medical students' attitudes towards the use of mathematical modelling as a means to improve research skills

Perry, Z. H., Todder, D. — Mon, 25 May 2009 06:46:14 PDT

A PUBMED search for ‘mathematical models in medicine’ shows more than 15,000 articles covering almost every field of medicine. We designed a course with the goal of developing the students’ skills in computerized data analysis and mathematical modelling, as well as enhancing their ability to read and interpret mathematical data analysis. The study evaluated the acquisition of research skills and how to understand such data, as well evaluating the students’ feeling of competence. The course was structured as a 1-week (30-h) workshop for final year medical students. The study population consisted of 23 medical students who took the course in the 2005 academic year. Course evaluation used questionnaires that assessed the students’ satisfaction and mathematical knowledge. We found a significant change in the attitudes of our subjects, comparing their before and after attitudes towards their competence in the use of mathematical modelling, academically (i.e. their ability to read and understand articles using math models) as well as medically (i.e. their ability to implement theory that arises from math models to medical applications). We believe that the use of math modelling training in medical education significantly improved the students’ confidence in reading and applying math models in medicine; there is a tendency (albeit insignificant) towards superior results in attitudes of students towards math usage in medicine at large.

Solving second-order ordinary differential equations without using complex numbers

Kougias, I. E. — Mon, 25 May 2009 06:46:14 PDT

Ordinary differential equations (ODEs) is a subject with a wide range of applications and the need of introducing it to students often arises in the last year of high school, as well as in the early stages of tertiary education. The usual methods of solving second-order ODEs with constant coefficients, among others, rely upon the use of complex variable analysis, a topic to which the students in question may not have been previously introduced. The purpose of this article is to present an alternative approach in establishing the general solution for such types of equations without using complex numbers.

Khayyam with Cabri: experiences of pre-service mathematics teachers with Khayyam's solution of cubic equations in dynamic geometry environment

Baki, A., Guven, B. — Fri, 27 Feb 2009 05:46:44 PST

The study reported in this article deals with the observed actions of Turkish pre-service mathematics teachers in dynamic geometry environment (DGE) as they were learning Khayyam's method for solving cubic equations formed as x³ + ax = b. Having learned the method, modelled it in DGE and verified the correctness of the solution, students generated their own methods for solving different types of cubic equations such as x³ + ax² = b and x³ + a = bx in the light of Khayyam's method. With the presented teaching experiment, students realized that Khayyam's mathematics is different from theirs. We consider that this gave them an opportunity to have an insight about the cultural and social aspects of mathematics. In addition, the teaching experiment showed that dynamic geometry software is an excellent tool for doing mathematics because of their dynamic nature and accurate constructions. And, it can be easily concluded that the history of mathematics is useful resource for enriching mathematics learning environment.

HMS--harmonic motion by shadows

Glaister, P., Glaister, E. M. — Fri, 27 Feb 2009 05:46:44 PST

A problem is discussed which is generated by shadows and which is a generalization of simple harmonic motion.

On Feynman's Triangle problem and the Routh Theorem

Man, Y.-K. — Fri, 27 Feb 2009 05:46:44 PST

In this article, we give a brief history of the Feynman's Triangle problem and describe a simple method to solve a general version of this problem, which is called the Routh Theorem. This method could be found useful to school teachers, instructors or lecturers who are involved in teaching geometry.

Open-start mathematics problems: an approach to assessing problem solving

Monaghan, J., Pool, P., Roper, T., Threlfall, J. — Fri, 27 Feb 2009 05:46:44 PST

This article describes one type of mathematical problem, open-start problems, and discusses their potential for use in assessment. In open-start problems how one starts to address the problem can vary but they have a correct answer. We argue that the use of open-start problems in assessment could positively influence classroom mathematics teaching. The article provides a brief review of problem solving and describes open-start problems in detail. The article then considers how open-start problems could address some important concerns in the teaching and assessment of mathematics and raises issues with regard to the future use of open-start problems in assessment.

Enlisting excel--again

Parramore, K. — Fri, 27 Feb 2009 05:46:44 PST

In Volume 26, Number 2, we reported on a group case study run for level 3 mathematics students at the University of Brighton. At the core of the study was a quadratic assignment problem, and we reported on attempts by students to use Excel to solve the problem, and on the attendant difficulties. We provided an elegant solution. In this article, we report on another interesting case study, this time developed for level 2 mathematics students. Again Excel is used to achieve a solution. We report on our efforts—and on how student answers led to improvements.

Minimizing the delay at traffic lights

Van Hecke, T. — Fri, 27 Feb 2009 05:46:44 PST

Vehicles holding at traffic lights is a typical queuing problem. At crossings the vehicles experience delay in both directions. Longer periods with green lights in one direction are disadvantageous for the vehicles coming from the other direction. The total delay for getting through the traffic point is what counts. This article presents an expression to calculate the optimal time periods of red lights and green lights starting from a fixed-cycle time. The solution is optimal if it makes the traffic jam delay at the road crossing minimal. As these solutions depend on the number of cars arriving in the different directions, which is not constant during the day, the application can be enlarged to a system where the time periods of red and green lights change during the day.

A study of two-term unit fraction expansions via geometric approach

Man, Y.-K. — Fri, 27 Feb 2009 05:46:44 PST

In this article, we report a study of two-term unit fraction expansions using a geometric approach. It provides us insight on how to find all two-term unit fraction expansions and identify the property of the expansions with smallest maximal denominators, for a given unit fraction. These findings will be useful to lecturers or teachers involved in teaching history of mathematics or elementary number theory.

The wonky trammel of Archimedes

Sangwin, C. — Fri, 27 Feb 2009 05:46:44 PST

This article examines the ellipsograph of Archimedes, also known as the locus problem of Franciscus van Schooten, and related mechanisms. We not only solve the algebraic system explicitly, but we also reverse engineer the problem and find configurations that provide a particular solution. Using the modern techniques of polynomial Gröbner Bases we show this can be used as a traditional compass, as a straight edge (to draw a straight line) and as an ellipsograph to trace ellipses.