Teaching Mathematics and its Applications - recent issues

Bridging the gap between mathematical conjecture and proof through computer-supported cognitive conflicts

Lee, C.-Y., Chen, M.-P. — 2008-02-18

In many mathematical problems, students can feel that the universality of a conjecture or a formula is validated by their experiment and experience. In contrast, students generally do not feel that deductive explanations strengthen their conviction that a conjecture or a formula is true. In order to cope up with students’ conviction based only on empirical experience and to create a need for deductive explanations, we developed a problem-solving activity with technology support intended to cause cognitive conflict. In this article, we describe the process conducted for this activity that led students to contradictions between conjectures and findings. The teacher could create familiar problem-solving situations and use students’ naïve inductive approaches to make students think mathematically and establish the necessity for proof via computer support.

Reorganizing freshman business mathematics I: background and philosophy

Green, K., Emerson, A. — 2008-02-18

This article is the first of the two-part discussion of the development of a new Freshman Business Mathematics (FBM) course at our college. Part I of the article describes the background and history behind the course, and provides a theoretical framework for the design of the course. This design involves students in learning and applying mathematics to real world problems using common business tools, in this case spreadsheets. This new course is centered on the concept of building and interpreting models of data, but touches on many topics in statistics, pre-calculus and calculus.

Using dynamic geometry software to convey real-world situations into the classroom: the experience of student mathematics teachers with a minimum network problem

Guven, B. — 2008-02-18

As any ordinary person knows, the shortest distance between two points is a straight line. What, then, is the shortest distance between three points? Four points? The study reported in this article deals with the observed actions of Turkish student mathematics teachers as they were working with minimal network problems. Having analysed the mathematization processes of student mathematics teachers in computerized environment, I describe here how new mathematical relationships can be discovered from real-world situations. The results showed that using real world situations in computerized classrooms leaves the doors open for the students for decision making, experimental verification, conjecturing and even for construction of proofs.

Complex variables in junior high school: the role and potential impact of an outreach mathematician

Duke, B. J., Dwyer, J. F., Wilhelm, J., Moskal, B. — 2008-02-18

Outreach mathematicians are college faculty who are trained in mathematics but who undertake an active role in improving primary and secondary education. This role is examined through a study where an outreach mathematician introduced the concept of complex variables to junior high school students in the United States with the goal of stimulating their interest in mathematics and improving their algebra skills. Comparison of pre- and post-test results showed that ninth-grade students displayed a significant change in algebraic skills while the eighth-grade students made little progress. The outreach mathematician lacked some awareness of the eighth-grade students’ foundational background and motivation. This illustrates the importance of working more closely with the participating teacher, who understands better the curriculum and the students’ background knowledge, levels of maturity and levels of motivation.

Getting the best out of Excel

Heys, C. — 2008-02-18

Excel, Microsoft's spreadsheet program, offers several tools which have proven useful in solving some optimization problems that arise in operations research.

We will look at two such tools, the Excel modules called Solver and Goal Seek—this after deriving an equation, called the ‘cash accumulation equation’, to be used in conjunction with them.

Editorial note

2007-11-28

Retention and progression of engineering students with diverse mathematical backgrounds

Bamforth, S. E., Robinson, C. L., Croft, T., Crawford, A. — 2007-11-28

There are increasing concerns about the mathematics ability of students entering higher education. This situation appears to be as a result of the perceived lowering standards of A Levels, a reduction in entry requirements on some courses with a strong mathematical component and the wide-ranging educational backgrounds of many of the students.

With Additional Student Numbers (ASN) funding, a pre-sessional course has been introduced at Loughborough University as a collaboration between the Engineering Centre for Excellence in Teaching and Learning (engCETL) and the Mathematics Education Centre (MEC), also a designated Centre for Excellence. The 4 day, residential, pre-sessional course targets engineering students with diverse mathematical backgrounds just before the start of the first year of their degree course. Due to possible gaps in their mathematical knowledge, students with non-traditional mathematics backgrounds are at risk of struggling in traditional lectures where a certain level of knowledge is assumed. The pre-sessional course, called Flying Start, aims both to reinforce the need for mathematical competency and to raise awareness of and to encourage students to engage with the support facilities available to them once they start at University. The main constituents of the Flying Start course are mathematics and engineering key skills workshops. Flying Start was introduced in 2003, growing from a pilot of 11 Electrical Engineering students to 24 students from Electrical, Manufacturing and Materials Engineering in 2005. Following each course, the performance of the Electrical Engineering students is monitored throughout the first year. This article examines the Flying Start students’ academic performance in light of their mathematical background and their uptake of the additional support on offer. Student feedback suggests that the pre-sessional course offers the additional benefit of aiding the students in their transition into higher education. The implications of the feedback and the student performance data are also discussed.

Musing on the use of dynamic software and mathematics epistemology

Santos-Trigo, M., Reyes-Rodriguez, A., Espinosa-Perez, H. — 2007-11-28

Different computational tools may offer teachers and students distinct opportunities in representing, exploring and solving mathematical tasks. In this context, we illustrate that the use of dynamic software (Cabri Geometry) helped high school teachers to think of and represent a particular task dynamically. In this process, the teachers had the opportunity of identifying, exploring and supporting mathematical relations that emerged during the solution of the task. We distinguish problem-solving episodes that teachers exhibited while understanding and representing the task, thinking of a solution plan, searching and presenting mathematical arguments and looking for connections.

The Towers of Hanoi

Morris, G. C. — 2007-11-28

This article presents an investigation carried out with a group of able mathematics students who were studying at a level 1 year in advance of their peers. The purpose was to investigate the extension of usual three peg Towers of Hanoi to four pegs and attempt to find a rule that could be used to predict the minimum number of moves required to complete the puzzle.

Does a cube have an equation?

Satianov, P., Fried, M. N. — 2007-11-28

This article looks at the question of whether and how a geometrical cube may be determined as the solution set of a single equation. Beyond this being an interesting and surprising result for upper high school and first year college students, we argue that, as an investigative activity, it has value in refining students’ images of what solutions of equations can be and, with that, refining their ideas of what an equation is.

ODE without complex numbers: another look

Dobos, J. — 2007-11-28

The aim of this paper is to present a simple way of establishing the general solution of the linear, second-order differential equation without using complex numbers.

Geometrical analogies in mathematics lessons

Eid, W. — 2007-11-28

A typical form of thinking to approach problem solutions humanly is thinking in analogous structures. Therefore school, especially mathematical lessons should help to form and to develop corresponding heuristic abilities of the pupils. In the contribution, a summary of possibilities of mathematics lessons regarding this shall particularly be conveyed from the view of geometry lessons in different age groups, being supposed to be exemplarily illustrated introducing character and object of making analogies as an example. Different kinds of analogous conclusions are described exemplarily with some examples for different geometrical contents and didactic situations.

Spidergraph

Glaister, P., Glaister, E. M., Glaister, A. E., Glaister, M. A. — 2007-11-28

Some interesting and unusual loci are generated by a familiar mechanical system.

Poker Face

Fletcher, M. — 2007-11-28

Preface

Hibberd, S., Mustoe, L. — 2007-09-12

Recent changes in A-level Mathematics: is the availability and uptake of mechanics declining yet more?

Lee, S., Harrison, M. C., Robinson, C. L. — 2007-09-12

In the past 6 years changes have occurred in GCE A-levels. In particular, there have been several major changes in A-level Mathematics courses. As engineering students are usually required to have studied A-level Mathematics, or its equivalent, these changes have had an effect on their prior mathematical knowledge. Moreover, engineering students traditionally obtained a good grounding in mechanics as part of their A-level Mathematics qualification. However, mechanics, which was once included in the core syllabus, is now optional. This article investigates the current availability and uptake of mechanics modules within A-level Mathematics courses in schools. Comparisons are drawn between these results and results of a survey of schools in 2004 and surveys of first year engineering students conducted in 2004 and 2005. It is found that there is a decline in the availability of mechanics modules and the uptake of more than one mechanics module has also decreased. The implications of these findings for engineering educators are discussed.

Prior knowledge of mechanics amongst first year Engineering students

Clements, D. — 2007-09-12

In the last 25 years, A-level Mathematics syllabi have changed very considerably, introducing a broader range of application areas but reducing the previous emphasis on classical mechanics. This article describes a baseline survey undertaken to establish in detail the entry levels in mechanics for the cohort of students entering Engineering courses at Bristol University in October 2005. The survey results confirm and quantify existing anecdotal evidence indicating that universities must now assume a considerably reduced familiarity with concepts in basic mechanics. These changes have strong implications for future course design.

Computer-aided assessment in mechanics: question design and test evaluation

Gill, M., Greenhow, M. — 2007-09-12

This article describes pedagogic issues in setting objective tests in mechanics using Question Mark Perception, coupled with MathML mathematics mark-up and the Scalable Vector Graphics (SVG) syntax for producing diagrams. The content of the questions (for a range of question types such as multi-choice, numerical input and variants such as confidence-based questions) is scripted with random parameters, thereby producing many millions of realizations of the underlying ‘question style’. This means that the question setter must completely specify the algebraic and pedagogic structure of the question. For some question types, we need to understand and encode the ways in which students make mistakes, offering them as distracters or recognizing their use in numerical inputs (we call this responsive numerical input). We have examined several years’ worth of exam scripts to discover what ‘mal-rules’ are used for each question and attempted to characterize them with metadata that makes students’ responses recorded in the answer files easier to understand.

Results from evaluation experiments are presented; in particular, we are interested in whether the feedback ‘feeds forward’ to affect students’ approaches to doing problems in a repeat test or exam, delayed by a variable time period (almost immediately, after 1 week, 1 month or more). To quantify this when examining end of semester exam scripts, we looked at four indicators: using units, identifying vectors, using diagrams and emulating the good layout of the feedback screens in their own written solutions.

The effectiveness of support for students with non-traditional mathematics backgrounds

Symonds, R. J., Lawson, D. A., Robinson, C. L. — 2007-09-12

This article describes an initiative introduced at Loughborough University by SIGMA, a Centre for Excellence in Teaching and Learning (CETL), to support physics students who were mathematically less well-prepared than their counterparts. The article outlines how students were identified as being less well-prepared. These students were taught in a separate group, using different materials and a different teaching style, but the same assessment methods were used for both groups. An evaluation of the success of this initiative is made by comparing the results of the less well-prepared students receiving support in 2005–06 with those of the less well-prepared students (taught in the mainstream group) in 2004–05. A key outcome of this comparison is an increase in the pass rate from 48% to 67%.

Observing mathematics teaching in three different countries lessons learnt?

Metje, N. — 2007-09-12

It has been reported that the students’ level of mathematics when entering university has changed in recent years. A surprising number of students struggle with mathematics or have a mathematics anxiety. Although this was widely reported within the UK, there was some uncertainty if colleagues in other countries were experiencing the same challenges. Therefore, lectures in three countries were observed and the effects different educational backgrounds and cultures have on the students’ attitudes towards mathematics were determined. Experiences and ideas were exchanged and it became clear that the problems lecturers face in the UK with respect to teaching mathematics are in fact not limited to this country but that their approach to lecturing at university is different.

This article discusses the outcomes of analysing and observing mathematics taught in three different countries. Similarities and differences are presented, together with feedback on the success of implementing new ideas at Birmingham.