Teaching Mathematics and its Applications - recent issues

Does students' confidence in their ability in mathematics matter?

Parsons, S., Croft, T., Harrison, M. — 2009-05-25

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. — 2009-05-25

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. — 2009-05-25

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. — 2009-05-25

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. — 2009-05-25

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. — 2009-02-27

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. — 2009-02-27

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. — 2009-02-27

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. — 2009-02-27

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. — 2009-02-27

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. — 2009-02-27

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. — 2009-02-27

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. — 2009-02-27

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.

Mathematics support--support for all?

Pell, G., Croft, T. — 2008-11-18

Mathematics Support Centres are to be found in various forms in the majority of UK higher education institutions. They have been established in order to ease widespread and serious difficulties that a significant number of students have with mathematics, particularly at the school–university transition. They usually offer mathematics and/or statistics support to students across the full range of disciplines studied. Anecdotal evidence suggests that those students who make good use of such centres are not just those who struggle with mathematics. Many frequent users are quite competent and simply want to do better. The study reported here describes and analyses data from one cohort of engineering students. A novel aspect is the quantification of the proportion of support centre visitors who fall into these, and other, categories. We conclude of the cohort in the study, mathematics support has improved the pass rate by ~3%. Of the failures, about half (~4% of the sample total) could well have passed had they attended the mathematics support centre regularly. Furthermore, the majority of those attending were not students who were in danger of failing. This has important implications not only for the design of mathematics support provision, but also for the performance of the high fliers. The methodology offers one way tackling the difficult task of evaluating the effectiveness of mathematics support initiatives.

Rolling Squares

Holton, D., Knights, C. — 2008-11-18

Here, we investigate what loci are produced when a square of side-length one is allowed to rotate around a square of side-length n, where n is a whole number. We find that if i = 1, 2, 3 or 4 (mod 4), the loci obtained for n i (mod 4) all have the same symmetry and we show how the perimeter of each class can be determined. We also analyse the case of rotation of a square of side-length one about a rectangle of dimensions m x n when m and n are whole numbers. A similar result is obtained for classes of rectangles modulo 4 to that obtained in the square case. Finally, we suggest that this is a useful investigation to undertake with secondary school pupils. We discuss how we have used this investigation with such a group.

Why mechanics should be integral to secondary school mathematics

Rowlands, S. — 2008-11-18

Mechanics has never been the most popular subject in A-level mathematics, the UK’s public examination for 16–18-year olds, either with students, teachers or educators. The attempts to popularize mechanics have failed and it is conceivable that the subject will be dropped from the A-level syllabus in the foreseeable future. This article argues the importance of mechanics and why it should be integral to secondary school mathematics: Mechanics is the exemplar of mathematical modelling, is the logical point of entry for the enculturation into scientific thinking and provides the means to develop an understanding of the relationship between mathematics, the theoretical objects of science and the way science and mathematics speak of the world. It enables learners across the ‘ability range’ to think in the abstract and as such should be taught prior to the 6th form, that is, prior to the UK’s post-compulsory level of education.

Pythagorean approximations and continued fractions

Peralta, J. — 2008-11-18

In this article, we will show that the Pythagorean approximations of $$\sqrt{2}$$ coincide with those achieved in the 16th century by means of continued fractions. Assuming this fact and the known relation that connects the Fibonacci sequence with the golden section, we shall establish a procedure to obtain sequences of rational numbers converging to different algebraic irrationals. We will see how approximations to some irrational numbers, using known facts from the history of mathematics, may perhaps help to acquire a better comprehension of the real numbers and their properties at further mathematics level.

Showing you're working: a project using former pupils' experiences to engage current mathematics students

Musto, G. — 2008-11-18

To help students view mathematics in a more favourable light, a number of former pupils were contacted and asked to give details of how they use mathematics in their daily lives. This information was gathered through an online questionnaire or visits to the school to talk to pupils—a booklet of responses was also given to students. Attitudinally pre- and post-testing students suggested that this initiative helped address pupils’ concerns regarding the purpose of classroom mathematics. The diversity of professions also helped dispel many myths about the usefulness of mathematics. Subsequently, the project has proven to be a catalyst for a range of cross-curricular projects and events inspired by the former pupils’ case studies, all of which serve to continue to address the initial aims of the project regarding pupil perception of the subject, in the light of both workplace and everyday life.

A short and elementary proof of the infinitude of primes

Scimone, A. — 2008-11-18

Proof without words: Formula

Khattri, S. K. — 2008-11-18

We present a pictorial proof of the inequation $$(1+\frac{1}{n}){}^{n} < e < (1+\frac{1}{n}){}^{n+1}$$. The inequation is also confirmed through the Taylor expansion and alternating series theorem.