Showing papers in "International Journal of Mathematical Education in Science and Technology in 2003"

The measurement of school students' understanding of statistical variation

Abstract: The paper presents a questionnaire devised to assess school students' understanding of statistical variation. The questionnaire is based on earlier research into students' understanding of the chance and data curriculum and recent work, more specifically related to variation. It was devised, piloted, revised, and administered to 746 students in grades 3, 5, 7, and 9 in ten Tasmanian schools. The analysis of outcomes was carried out in three stages: a hierarchical coding scheme was developed based on a structural model of cognitive development; a Rasch analysis was carried out to produce a variable map of student performance and item difficulty on a single scale; and a holistic model of development was suggested for the questionnaire. Outcomes for individual items are presented to illustrate the range of student responses, and possible rubrics for use by teachers. For some items, comparisons are made with results of other researchers.

Singaporean students' mathematical thinking in problem solving and problem posing: an exploratory study

Abstract: This study explored Singaporean fourth, fifth, and sixth grade students' mathematical thinking in problem solving and problem posing. The results of this study showed that the majority of Singaporean fourth, fifth, and sixth graders are able to select appropriate solution strategies to solve these problems, and choose appropriate solution representations to clearly communicate their solution processes. Most Singaporean students are able to pose problems beyond the initial figures in the pattern. The results of this study also showed that across the four tasks, as the grade level advances, a higher percentage of students in that grade level show evidence of having correct answers. Surprisingly, the overall statistically significant differences across the three grade levels are mainly due to statistically significant differences between fourth and fifth grade students. Between fifth and sixth grade students, there are no statistically significant differences in most of the analyses. Compared to the findings...

The concept of rhotrix in mathematical enrichment

Abstract: This paper introduces the concept of rhotrices, their algebra and analysis, and presents them for mathematical consideration.

New opportunities for encouraging higher level mathematical learning by creative use of emerging computer aided assessment

Abstract: This article defines ‘higher level mathematical skills’ and details an important class: that of constructing instances of mathematical objects satisfying certain properties. Comment is made on the frequency of higher level tasks in undergraduate work. We explain how such questions may be assessed in practice without the imposition on staff of an onerous marking load. Included are examples which have been implemented on a free computer aided assessment system. Lastly we report an investigation of students’ reactions to these questions and discuss their design and impact.

Engendering proof attitudes: can the genesis of mathematical knowledge teach us anything?

Abstract: The necessity for reading and writing proofs is one of the central distinguishing characteristics of university mathematics. But reports over the last decade voice concern about students' understanding of mathematical proof at both secondary school and undergraduate level. These reports suggest that there is continued poverty in the understanding of proof by students. This article reports on how proof attitudes could be inculcated in students by offering them a course design that is faithful, to some extent, to the historical genesis of modern mathematics. Specifically, it is argued that such a design can enable students to discover for themselves a ‘sense of proof’.

An analysis of undergraduate core material in the light of hand-held computer algebra systems

Abstract: This study investigates how hand-held computer algebra systems may be used to complete undergraduate mathematics course work tasks. Core university mathematics courses are examined in detail using a question classification scheme that describes the style of a question, rather than the syllabus topic. The majority of the tasks examined could be completed directly using the device. Surreptitious use of the hand-held device would subvert the intentions of the course designer.

Classroom note: Application of the Sumudu transform to discrete dynamic systems

Abstract: The Sumudu transform is an integral transform introduced to solve differential equations and control engineering problems. The transform possesses many interesting properties that make visualization easier and application has been demonstrated in the solution of partial differential equations, integral equations, integro-differential equations and dynamic/dynamic-differential systems. In this note, application of the method of Sumudu transform to the solution of discrete dynamic systems is described. Derivations find ready application in recurrence relations. Solutions are obtained to certain first degree, second degree and third degree dynamic systems.

Projecting Rate of Change in the Context of Motion onto the Context of Money

Abstract: This paper reports a study designed to probe the abstraction ability that US Algebra I students can achieve in moving the concepts rate of change and accumulation between motion and money contexts. The Algebra I students used the technologies motion detectors and Interactive Banking software during a replacement unit focused on slope, ratio, and rate of change. Clinical interviews were conducted with four students at the end of the replacement unit testing their abstraction ability across the two contexts. Results revealed that some students do not have to completely understand the relationship between rate of change and accumulation within a single context in order to be able to understand and project the concepts separately into multiple contexts. Using multiple rate of change contexts allowed the learners the opportunity to see the ‘like’ in the contextually unlike situation, enabling them to project these concepts into novel situations. Few research studies have examined students’ understanding of rat...

Coupled spring equations

Abstract: Coupled spring equations for modelling the motion of two springs with weights attached, hung in series from the ceiling are described. For the linear model using Hooke's Law, the motion of each weight is described by a fourth- order linear differential equation. A nonlinear model is also described and damping and external forcing are considered. The model has many features that permit the meaningful introduction of many concepts including: accuracy of numerical algorithms, dependence on parameters and initial conditions, phase and synchronization, periodicity, beats, linear and nonlinear resonance, limit cycles, harmonic and subharmonic solutions. These solutions produce a wide variety of interesting motions and the model is suitable for study as a computer laboratory project in a beginning course on differential equations or as an individual or a small-groupundergraduate research p roject.

Revealing hidden mathematics curriculum to pre-teachers using technology: the case of partitions

Abstract: Motivated by work done with pre-teachers of mathematics in a problem-solving course, this paper shows how computing technologies, including a spreadsheet and Maple, facilitate an informal journey into a hidden aspect of the formal content of the pre-college curriculum dealing with the arithmetic of partitions. By using three problems from different grade levels within a state curriculum as an example, the paper suggests that a deeper perspective on seemingly disconnected problem-solving contexts may serve as a powerful didactical tool in helping teachers to appreciate mathematics and its pedagogy as an integrated whole. The connection of the hidden aspect of the curriculum to the concept of mathematical play is also explored.

Some combinatorics of multisets

Abstract: The purpose of this paper is to delineate some insights into the concept of multisets along with a couple of combinatorial results related to multisets. The paper indicates that a general formula needs to be worked out for determining the cardinality of the ‘Set’ (whose elements may be multisets but do not repeat) of all multisubsets of a finite multiset [x, y, z,…] m , p , t , … in which x occurs at most m times, y occurs at most p times, and so on. It outlines some directions provided in the literature and points out that they all turn out to be inefficient. Finally, a relatively more efficient formula to this effect is provided along with a remark that the problem needs further vindication.

The Impact of Beliefs in Student's Learning: An Investigation with Students of Two Different Contexts.

Abstract: In the last decade, the community of mathematics educators has increased the attention paid to teachers' and students' beliefs, given their impact on the learning of mathematics. This paper focuses on two secondary students' beliefs and the relationships that emerge from the actions that take place in a classroom. The study was conducted in two different countries, Spain and Costa Rica, following an interpretative approach. It has been found that students construct their own theories regarding their role at school and learning and teaching mathematics. Both students agree that school mathematics is typically based on rules and memorization, and mostly oriented to procedures; even though they had contrasting mathematics experiences at school, their views about issues discussed in this paper are similar.

Investigating students' approval of a geometrical approach to differential equations and their solutions

Abstract: The advancement of computer graphics in the past decade has contributed to an enormous change in the way mathematics is taught at the college level. Although Calculus has been the main benefactor of this advancement, the topic of ordinary differential equations has undergone fundamental changes in its curriculum in favour of the visual and numerical aspects of the course. This paper investigates students' acceptance of solving a differential equation geometrically. The setting is an introductory course on ordinary differential equations offered to engineering students, in which a reformed approach was adopted. Data were gathered from exams, and interviews were conducted with six students who volunteered to take part in the study. Results show that students initially showed reluctance to accept this geometric approach, but many of them eventually accepted it, appreciated its usefulness, and wished that other math courses be offered in a similar way.

The Use of Graphics Calculators by Students in an Examination: What Do They Really Do?

Abstract: In many British schools, A-level Mathematics students are advised to purchase graphics calculators at the start of their Mathematics course, and there has been considerable research into the effectiveness of the use of graphics calculators in developing mathematical understanding. Recent UK examination regulations have prohibited the use of graphics calculators in certain module examinations but allowed them to be used in others. This study set out to investigate how a small group of students actually used their graphics calculators under examination conditions. The students sat an externally set practice examination paper for a statistics module. The examination paper was analysed by the research team in order to identify the potential use that the students could have made of the graphics calculators in each question. When they took the examination the students were provided with specially adapted calculators; these calculators were virtually identical to the students' own calculators but contained speci...

A three-year perspective on conceptions of and orientations to learning mathematics of prospective teachers and first year university students

Abstract: The paper reports a compilation of results from three studies conducted over three years to determine students' conceptions of mathematics, and orientations they follow in learning the subject. Respondents were 459 first year mathematics students from four universities and one teacher college. Results indicated that more than half the sample reported mathematics to be a subject made of numbers and formulae that could be memorized. This suggests a shallow emphasis when learning the subject, with no intention to understand. However, most students passed their examinations. It was concluded that there was no statistically significant relationship between examinations results and students' learning orientations. It is recommended that lecturers should foster students' meta-learning capabilities and an awareness of different learning strategies.

On the role of kinesthetic thinking in computational geometry

Abstract: Computational geometry is a new (about 30 years) and rapidly growing branch of knowledge in computer science that deals with the analysis and design of algorithms for solving geometric problems These problems typically arise in computer graphics, image processing, computer vision, robotics, manufacturing, knot theory, polymer physics and molecular biology Since its inception many of the algorithms proposed for solving geometric problems, published in the literature, have been found to be incorrect These incorrect algorithms rather than being ‘purely mathematical’ often contain a strong kinesthetic component This paper explores the relationship between computational geometric thinking and kinesthetic thinking, the effect of the latter on the correctness and efficiency of the resulting algorithms, and their implications for education

Flexible on-line assessment and feedback for teaching linear algebra

Abstract: There is an ongoing demand for feedback by students, and this in turn creates a need for assessment support. In this respect, the use of on-line assessment has a number of obvious advantages for both the instructor and the learner. Perhaps one of the most significant advantages is that it allows both of them to exploit existing technology to offer a more flexible learning environment. This paper reports on the use of an online assessment tool that provides interactive feedback to students learning linear algebra. The success is measured in terms of improved pass rate and students' satisfaction in the flexible learning opportunities that the tool offered.

The Gibbs' phenomenon for Fourier-Bessel series

Abstract: The paper investigates the Gibbs' phenomenon at a jump discontinuity for Fourier-Bessel series expansions. The unexpected thing is that the Gibbs' constant for Fourier-Bessel series appears to be the same as that for Fourier series expansions. In order to compute the coefficients for Fourier-Bessel functions efficiently, several integral formulas are derived and the Struve functions and their asymptotic expansions discussed, all of which significantly ease the computations. Three numerical examples are investigated. Findings suggest further investigations suitable for undergraduate research projects or small student group investigations.

A Generalization of Leibniz Rule for Higher Derivatives.

Abstract: This note provides a simple method to extend the usual Leibniz rule for higher derivatives of the product of two functions to several functions, which is within the reach of freshman calculus students.

Technology-active mathematical modelling

Abstract: 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

Mimicry of proofs with computers: the case of Linear Algebra

Abstract: This article examines the idea of 'following the flow of a proof with an example' in order to assist the learner in the challenging task of understanding mathematical proofs. This strategy is termed 'mimicry of a proof'. However, such mimicry can be impractical or unreasonably demanding when the mathematical objects in the proof are difficult to manipulate without technological enhancement. This is the case with many proofs in Linear Algebra, in which the manipulated objects are vectors or matrices. Therefore, the article focuses on the idea of proof mimicry with a computer algebra system (CAS). As examples, this strategy is applied to the proofs of two theorems: the basis theorem and the orthogonalization theorem. In addition, pedagogical guidelines to be followed in constructing a set of computer activities for students are presented and examined.

Computer derivations of numerical differentiation formulae

Abstract: Traditional ‘pencil and paper’ derivations of the numerical differentiation formulae for f′ [x 0] and f ″[x 0] have been done independently as if there was no connection among the two derivations. This new approach gives a parallel development of the formulae. It requires the solution of a 'linear system' that includes symbolic quantities as coefficients and constants. It is shown how the power of a computer algebra system such as Mathematica can be used to elegantly solve this linear system for f′[x 0] and f″[x 0]. The extension to derivations of higher order numerical differentiation formulas for the central, forward or backward differences are also presented. Traditional ‘pencil and paper’ derivations of the numerical differentiation formulae for f′ [x 0] and f ″[x 0] have been done independently as if there was no connection among the two derivations. This new approach gives a parallel development of the formulae. It requires the solution of a 'linear system' that includes symbolic quantities as coeff...

Classroom note: M-bonacci numbers and their finite sums

Abstract: The note considers M-bonacci numbers, which are a generalization of Fibonacci numbers. Two new summation formulas for M-bonacci numbers are given. The formulas are generalizations of the two summation formulas for Fibonacci numbers.

Bernoulli numbers via determinants

Abstract: In this note, Cramer's rule is used for the solution of a system of linear equations to represent Bernoulli numbers via determinants.

Regulation of a prey-predator fishery by taxation

Abstract: Bioeconomic exploitation of a prey-predator fishery is discussed by using a reasonable catch-rate function instead of the usual 'catch-per-unit-effort hypothesis'. The model is based on the assumption that only the prey species is harvested. A regulatory agency controls exploitation of the fishery by imposing a tax per unit biomass of the landed fish. The fishing effort is taken as a dynamic variable depending on the capital invested in the fishery. The dynamical system consisting of the growth equations of the species and also the fishing effort is formulated. The existence and stability of steady states are discussed. The optimal tax policy is studied with the help of Pontryagin's maximum principle. Finally, some numerical examples are carried out to illustrate the results of the model.

Inequalities and monotonicity of the ratio of the geometric means of a positive arithmetic sequence with unit difference

Abstract: For any nonnegative integer k and natural numbers n and m, the following inequalities are obtained on the ratio for the geometric means of a positive arithmetic sequence with unit difference: where α ∈ [0, 1] is a constant. Moreover, some monotonicity results for the sequences involving are obtained, and the related inequalities are generalized.

Secondary trainee teachers' understanding of convergence and continuity

Abstract: This paper explores the understanding which students who are training to be secondary mathematics teachers have of elementary concepts in mathematical analysis. Research to date has tended to concentrate on students' knowledge and understanding prior to university entrance or in the first year of undergraduate mathematics courses. The design of this research, similar in style to a viva voce, involved probing students' conceptual understanding of convergence and continuity and associated reasoning. The report tentatively concludes that even mathematically well qualified students have difficulties with some elementary concepts and that students with engineering backgrounds allow the 'dynamic' image to dominate their engagement with the concepts.

Nonlocal advection equations

Abstract: Nonlocal problems are largely ignored in graduate and undergraduate texts on partial differential equations. Yet, nonlocal advection equations are important in many applications, and their solution provides an excellent illustration of the method of characteristics and how careful one must be to apply it.

Microsoft Excel as a supplement in a high school statistics course

Abstract: This study was conducted in the spring of 2002 at Arlington High School in a Probability and Statistics course for junior and senior level students. Two sections of the class were compared with respect to the use/non-use of EXCEL in the course. The experimental group (the one using EXCEL) outperformed the control group on five of the six teacher-created unit tests and on both of the criterion referenced assessments used by the district; however, the differences were not statistically significant at the 0.05 level. Furthermore, the experimental group expressed very positive feelings towards the use of the software in a survey administered at the conclusion of the course.

A multidisciplinary graduate level project-based programme in applied statistics

Abstract: The use of a project-based approach for designing a one-year graduate level programme in applied statistics is addressed. The pedagogical approach, academic setting, and learning activities are described in a multidisciplinary context. Some comments on implementation of such a programme based on the results from five successive graduating classes at the University of Veracruz (1994–2000) are included.