A magnet spring model

The paper discusses an elementary spring model representing the motion of a magnet suspended from the ceiling at one end of a vertical spring which is held directly above a second magnet fixed on the floor. There are two cases depending upon the north–south pole orientation of the two magnets. The attraction or repelling force induced by the magnets follow an inverse quartic law and thus we are led to a nonlinear model suitable for discussion in a beginning differential equations course. Spring models are common fare in such courses, but usually only linear models with simple sinusoidal forcing are considered. The resultant model is autonomous and thus an energy approach permits a full phase portrait of the resultant motions in the phase plane. These phase portraits show interesting behaviour of the system, reinforcing one's natural physical intuition. The computer algebra system Mathematica is employed here, although almost any other system would suffice. Such a system permits almost effortless calculations and can generate the graphics needed to thoroughly investigate the model.     

The Mathematics Attitudes and Perceptions Survey: an instrument to assess expert-like views and dispositions among undergraduate mathematics students

One goal of an undergraduate education in mathematics is to help students develop a productive disposition towards mathematics. A way of conceiving of this is as helping mathematical novices transition to more expert-like perceptions of mathematics. This conceptualization creates a need for a way to characterize students' perceptions of mathematics in authentic educational settings. This article presents a survey, the Mathematics Attitudes and Perceptions Survey (MAPS), designed to address this need. We present the development of the MAPS instrument and its validation on a large (N = 3411) set of student data. Results from various MAPS implementations corroborate results from analogous instruments in other STEM disciplines. We present these results and highlight some in particular: MAPS scores correlate with course grades; students tend to move away from expert-like orientations over a semester or year of taking a mathematics course; and interactive-engagement type lectures have less of a negative impact, but no positive impact, on students' overall orientations than traditional lecturing. We include the MAPS instrument in this article and suggest ways in which it may deepen our understanding of undergraduate mathematics education.     

Factors influencing students’ perceptions of their quantitative skills

There is international agreement that quantitative skills (QS) are an essential graduate competence in science. QS refer to the application of mathematical and statistical thinking and reasoning in science. This study reports on the use of the Science Students Skills Inventory to capture final year science students’ perceptions of their QS across multiple indicators, at two Australian research-intensive universities. Statistical analysis reveals several variables predicting higher levels of self-rated competence in QS: students’ grade point average, students’ perceptions of inclusion of QS in the science degree programme, their confidence in QS, and their belief that QS will be useful in the future. The findings are discussed in terms of implications for designing science curricula more effectively to build students’ QS throughout science degree programmes. Suggestions for further research are offered.     

Policy and the standards debate: mapping changes in assessment in mathematics

ABSTRACT

The influences on governments for policy changes in schools range across many agencies, including the political party in power. When policies change, the sources of these influences are not always clear. The project whose work is presented in this special issue examines what these changes look like in terms of the differences in assessment tasks of school pupils’ mathematics, over time. In this article we attempt to develop a graph, which we argue will have general applicability internationally, that can help to reveal the sources and nature of those influences. We construct the graph in interaction with an examination of the most recent changes in two countries. We argue that our analysis is a necessary complement to the project’s findings in that it enables us to identify the fields of recontextualisation, their relative strengths in terms of influence and hence conjecture their impact on the mathematics curriculum.     

A model of students’ combinatorial thinking

Combinatorial topics have become increasingly prevalent in K-12 and undergraduate curricula, yet research on combinatorics education indicates that students face difficulties when solving counting problems. The research community has not yet addressed students’ ways of thinking at a level that facilitates deeper understanding of how students conceptualize counting problems. To this end, a model of students’ combinatorial thinking was empirically and theoretically developed; it represents a conceptual analysis of students’ thinking related to counting and has been refined through analyzing students’ counting activity. In this paper, the model is presented, and relationships between formulas/expressions, counting processes, and sets of outcomes are elaborated. Additionally, the usefulness and potential explanatory power of the model are demonstrated through examining data both from a study the author conducted, and from existing literature on combinatorics education.     

A mathematical model for unemployment

In this paper we have proposed and analyzed a non-linear mathematical model for unemployment by considering three variables, namely the numbers of unemployed, temporarily employed and regularly employed persons. The model is studied using the stability theory of differential equations. It is found that the model has only one equilibrium, which is non-linearly stable under certain conditions. Numerical simulation of the model has been carried out to confirm the analytical results.     

Developing flexible procedural knowledge in undergraduate calculus

Mathematics experts often choose appropriate procedures to produce an efficient or elegant solution to a mathematical task. This flexible procedural knowledge distinguishes novice and expert procedural performances. This article reports on an intervention intended to aid the development of undergraduate calculus students’ flexible use of procedures. Two sections of the same course were randomly assigned to treatment and control conditions. Treatment students completed an assignment on which they resolved derivative-finding problems with alternative methods and compared the two resulting solutions. Control students were assigned a list of functions to differentiate. On the post-intervention test, treatment students were more likely to use a variety of solution methods without prompting than the control. Moreover, the set of treatment section solutions were closer to those of a group of mathematics experts. This study presents evidence that not only is flexible procedural knowledge a key skill in tertiary mathematics, it can be taught.     

A framework to categorize students as learners based on their cognitive practices while learning differential equations and related concepts

There are numerous theories that offer cognitive processes of students of mathematics, all documenting various ways to describe knowledge acquisition leading to successful transitions from one stage to another, be it characterized by Dubinsky's encapsulation, Sfard's reification or Piaget's equilibration. We however are interested in the following question. Who succeeds at making the leap and can we describe the attributes that set them apart from the ones that do not? In this article, we offer a framework to categorize students as learners based on their individual approaches towards learning concepts in differential equations and related concepts – as demonstrated by their efforts to resolve a conflict, conserve and rebuild their cognitive structures.     

数学思想方法在高等数学教育中的作用

从高等数学教育改革的角度,阐述了数学思想方法的含义和高等数学中的基本数学思想方法,论述了在高等数学教育中加强数学思想方法教学的重要性.     

Teachers’ mathematical activity in inquiry-oriented instruction

This work investigates the relationship between teachers’ mathematical activity and the mathematical activity of their students. By analyzing the classroom video data of mathematicians implementing an inquiry-oriented abstract algebra curriculum I was able to identify a variety of ways in which teachers engaged in mathematical activity in response to the mathematical activity of their students. Further, my analysis considered the interactions between teachers’ mathematical activity and the mathematical activity of their students. This analysis suggests that teachers’ mathematical activity can play a significant role in supporting students’ mathematical development, in that it has the potential to both support students’ mathematical activity and influence the mathematical discourse of the classroom community.     

Conceptions of mathematics and student identity: implications for engineering education

Lecturers of first-year mathematics often have reason to believe that students enter university studies with naïve conceptions of mathematics and that more mature conceptions need to be developed in the classroom. Students’ conceptions of the nature and role of mathematics in current and future studies as well as future career are pedagogically important as they can impact on student learning and have the potential to influence how and what we teach. As part of ongoing longitudinal research into the experience of a cohort of students registered at the author's institution, students’ conceptions of mathematics were determined using a coding scheme developed elsewhere. In this article, I discuss how the cohort of students choosing to study engineering exhibits a view of mathematics as conceptual skill and as problem-solving, coherent with an accurate understanding of the role of mathematics in engineering. Parallel investigation shows, however, that the students do not embody designated identities as engineers.     

A mathematical model for continuous crystallization

J. Banasiak We discuss a mixed‐suspension, mixed‐product removal crystallizer operated at thermodynamic equilibrium. We derive and discuss the mathematical model based on population and mass balance equations and prove local existence and uniqueness of solutions using the method of characteristics. We also discuss the global existence of solutions for continuous and batch mode. Finally, a numerical simulation of a continuous crystallizer in steady state is presented. Copyright © 2015 John Wiley & Sons, Ltd.     

A mathematical model for asset pricing

Asset price dynamics is studied by using a system of ordinary differential equations which is derived by utilizing a new excess demand function introduced by Caginalp [4] for a market involving more information on demand and supply for a stock rather than their values at a particular price. Derivation is based on the finiteness of assets (rather than assuming unbounded arbitrage) in addition to investment strategies that are based on not only price momentum (trend) but also valuation considerations. For this new model and the older models which were extracted using the classical excess demand function by Caginalp and Balenovich [2] and [3], time evolutions of asset price are compared through numerical simulations.     

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.     

Explication as a lens for the formalization of mathematical theory through guided reinvention

Realistic Mathematics Education supports students’ formalization of their mathematical activity through guided reinvention. To operationalize “formalization” in a proof-oriented instructional context, I adapt Sjogren's (2010) claim that formal proof explicates (Carnap, 1950) informal proof. Explication means replacing unscientific or informal concepts with scientific ones. I use Carnap's criteria for successful explication – similarity, exactness, and fruitfulness – to demonstrate how the elements of mathematical theory – definitions, axioms, theorems, proofs – can each explicate their less formal correlates. This lens supports an express goal of the instructional project, which is to help students coordinate semantic (informal) and syntactic (formal) mathematical activity. I demonstrate the analytical value of the explication lens by applying it to examples of students’ mathematical activity drawn from a design experiment in undergraduate, neutral axiomatic geometry. I analyze the chains of meanings (Thompson, 2013) that emerged when formal elements were presented readymade alongside those emerging from guided reinvention.     

A choice option between proofs in linear algebra

At their final exam in linear algebra students at the author's university were given the possibility to choose between two types of proofs to be done. They could either prove two short statements by themselves or they could explain four steps in a given proof. This paper reports on investigations of students’ responses to the choice option compared to the final grade of their exam: the relation between choice and grade, students’ thoughts about why they preferred one proof to the other, and their opinions on having to choose. The analysis is based on the students’ solutions and answers in questionnaires distributed immediately after the exam. The paper concludes that the students’ opinions do not imply that there is a ‘best way’ of giving a proof, but a choice option is preferred by almost all students regardless of final grade. It is thus asserted that the choice-option-project should continue.     

Making a prescription drug

Many compounds have been developed to treat specific illnesses and conditions. However, in the past it was harder to analyse the interaction between compounds and drugs. But now we are able to state nonlinear medical models of several variables and find the right combinations of the compounds and drugs (that are showing promise) to treat a specific condition. We solve these problems with multi‐stage Monte Carlo optimization. An example is presented here along with a discussion of the scientific method and a new forecasting technique.     

Development of a mathematical ability test: a validity and reliability study

The identification of talented students accurately at an early age and the adaptation of the education provided to the students depending on their abilities are of great importance for the future of the countries. In this regard, this study aims to develop a mathematical ability test for the identification of the mathematical abilities of students and the determination of the relationships between the structure of abilities and these structures. Furthermore, this study adopts test development processes. A structure consisting of the factors of quantitative ability, causal ability, inductive/deductive reasoning ability, qualitative ability and spatial ability has been obtained following this study. The fit indices of the finalized version of the mathematical ability test of 24 items indicate the suitability of the test.     

A mathematical model for impulse resistance welding

We present a mathematical model of impulse resistance welding. It accounts for electrical, thermal and mechanical effects, which are non‐linearly coupled by the balance laws, constitutive equations and boundary conditions. The electrical effects of the weld machine are incorporated by a discrete oscillator circuit which is coupled to the field equations by a boundary condition. We prove the existence of weak solutions for a slightly simplified model which however still covers most of its essential features, e.g. the quadratic Joule heat term and a quadratic term due to non‐elastic energy dissipation. We discuss the numerical implementation in a 2D setting, present some numerical results and conclude with some remarks on future research. Copyright © 2003 John Wiley & Sons, Ltd.