The impact of instructor pedagogy on college calculus students’ attitude toward mathematics

College calculus teaches students important mathematical concepts and skills. The course also has a substantial impact on students’ attitude toward mathematics, affecting their career aspirations and desires to take more mathematics. This national US study of 3103 students at 123 colleges and universities tracks changes in students’ attitudes toward mathematics during a ‘mainstream’ calculus course while controlling for student backgrounds. The attitude measure combines students’ self-ratings of their mathematics confidence, interest in, and enjoyment of mathematics. Three major kinds of instructor pedagogy, identified through the factor analysis of 61 student-reported variables, are investigated for impact on student attitude as follows: (1) instructors who employ generally accepted ‘good teaching’ practices (e.g. clarity in presentation and answering questions, useful homework, fair exams, help outside of class) are found to have the most positive impact, particularly with students who began with a weaker initial attitude. (2) Use of educational ‘technology’ (e.g. graphing calculators, for demonstrations, in homework), on average, is found to have no impact on attitudes, except when used by graduate student instructors, which negatively affects students’ attitudes towards mathematics. (3) ‘Ambitious teaching’ (e.g. group work, word problems, ‘flipped’ reading, student explanations of thinking) has a small negative impact on student attitudes, while being a relatively more constructive influence only on students who already enjoyed a positive attitude toward mathematics and in classrooms with a large number of students. This study provides support for efforts to improve calculus teaching through the training of faculty and graduate students to use traditional ‘good teaching’ practices through professional development workshops and courses. As currently implemented, technology and ambitious pedagogical practices, while no doubt effective in certain classrooms, do not appear to have a reliable, positive impact on student attitudes toward mathematics.     

Mathematics capital in the educational field: Bourdieu and beyond

Mathematics education needs a better appreciation of the dominant power structures in the educational field: Bourdieu's theory of capital provides a good starting point. We argue from Bourdieu's perspective that school mathematics provides capital that is finely tuned to generationally reproduce the social structures that serve to keep the powerful in power, while ensuring that less powerful groups are led to accept their own failure in mathematics. Bourdieu's perspective thereby highlights theoretical inadequacies in much mathematics education research, insofar as it presumes a consensus about a ‘what works agenda’ for improving achievement for all. Drawing on one case where we manufactured awkward facts, we illustrate a Bourdieusian interpretation of mathematics capital as reproductive, and the crucial role of its cultural arbitrary. We then criticise the Bourdieusian concept of ‘mathematical capital’ as the value of mathematical competence in practice and propose to extend his tools to include the contradictory ‘use’ and ‘exchange’ values of mathematics instead: we will show how this conceptualisation goes ‘beyond Bourdieu’ and helps explain how teaching-learning might (ideally) produce ‘cultural use value’ in mathematical competence, while still recognising the contradictions teachers and learners face. Finally, we suggest how critical education research generally can benefit from this theoretical framework: (1) in exposing the interest of the dominant classes; but also (2) in researching critical pedagogic alternatives that challenge orthodoxy in educational policy and practice both in mathematics education and more generally.     

A signal-recovery system: asymptotic properties,and construction of an infinite-volume process

We consider a linear sequence of ‘nodes’, each of which can be in state 0 (‘off’) or 1 (‘on’). Signals from outside are sent to the rightmost node and travel instantaneously as far as possible to the left along nodes which are ‘on’. These nodes are immediately switched off, and become on again after a recovery time. The recovery times are independent exponentially distributed random variables. We present results for finite systems and use some of these results to construct an infinite-volume process (with signals ‘coming from infinity’), which has some peculiar properties. This construction is related to a question by Aldous and we hope that it sheds some light on, and stimulates further investigation of, that question.     

Investigating textbooks as crucial interfaces between culture, policy and teacher curricular practice: two contrasted case studies in France and Norway

This paper reports on an investigation of mathematics curriculum documents, commonly used textbooks and teacher ‘curricular practice’ with respect to educational traditions in France and Norway. The study has helped to develop a deeper understanding of (1) educational traditions in France and Norway; (2) the ways the educational traditions permeate the system, from policy documents through textbooks into the classroom; and (3) the connections between the worlds of policy, textbooks and teacher curricular practice in mathematics. Results show that French and Norwegian curricular documents and practices were influenced by egalitarian values, albeit differently interpreted and ‘lived’ in each country. In terms of mathematics, using a focus on grade 6 geometric transformations has shown that French textbooks and teacher curricular practices emphasized theoretical properties and mathematical reasoning, whereas the Norwegian counterparts stressed practical and inquiry-based activities. It is argued that mathematics textbooks can be regarded as a crucial interface between culture, policy and curricular practice, and as a pivotal resource in teachers’ resource system for curricular practice, even in times of digitization.     

A survey of OR utilization in Greece

A recent survey concerning OR utilization in firms and public organizations in Greece showed a rather poor profile. The survey covered 20 big organizations including the National Defense General Staff, national enterprises of public utilities, banks and firms from the textile, cement, oil, construction and other industries. Only a few of these organizations are using OR to a significant degree. The rest reported ‘poor’ or ‘no’ use of OR.Research/implementation problems most frequently appearing in public sector organizations include lack of data, insufficient qualified personnel and unfavorable attitude towards cooperation by affected parties. In private firms, the need for quick solutions and the use of OR analysts in line functions quite often imply the use of a ‘rule of thumb’ rather than a mathematical OR technique. In addition, OR study results quite often do not get implemented. One of the reasons is resistance by the affected parties, considering any proposal for reorganization put forward by other parties as a threat.The use of OR has proven beneficial for the using firm or organization. These benefits, however, are not as great as one might expect, due to the problems mentioned above. The future growth of OR utilization in Greece will depend on developments concerning the education of managers in OR methods, creation of OR groups in firms that will seek solutions to real-world, worth-while problems, publicizing successful OR applications, active support by top management, reforms in the educational system etc. The real push for OR utilization in Greece will come from drastic changes in administrative practices, which are directly related to the socioeconomic developments in the country.     

Outpatient appointment scheduling in presence of seasonal walk-ins

This study investigates appointment systems (AS), as combinations of access rules and appointment-scheduling rules, explicitly designed for dealing with walk-in seasonality. In terms of ‘access rules’, strategies are tested for adjusting capacity through intra-week, or monthly seasonality of walk-ins, or their combined effects. In terms of ‘appointment rules’, strategies are tested to determine which particular slots to double-book or leave open in cases where seasonal walk-in rates exceed or fall short of the overall yearly rate. In that regard, this study integrates capacity and appointment decisions, which are usually addressed in an isolated manner in previous studies. Simulation optimization is used to derive heuristic solutions to the appointment-scheduling problem, and the findings are compared in terms of in-clinic measures of patient wait time, physician idle time and overtime. The goal is to provide practical guidelines for healthcare practitioners on how to best design their AS when seasonal walk-ins exist.     

Historical comments on the use of technology and devices in ICMEs and ICMI

The ICMI Centennial in 2008 gave the occasion to study how the use of technology in teaching mathematics evolved during this period and was transformed by following the general development of technology. Given the key role of Felix Klein in the first international movement for reforming mathematics teaching and in particular for orienting it towards applications, his achievements in the use of geometrical models and artefacts are reported in this historical introduction. The term of ‘model’ proves to be essential in Klein’s efforts to restructure university teaching and to reform secondary school teaching.     

Multivariate analysis using ‘and’ and ‘or’

The connectives ‘and’ and ‘or’ are potentially useful in multivariate analysis and theory construction. They are simple, logical ways to connect two or more variables together. However, until recently there has been no framework for operationalizing these connectives for continuous variables, and this lack has severely limited their use. Using fuzzy set theory as a basis for such a framework, this paper lays out the necessary tools and models to permit the use of ‘and’ and ‘or’ in multivariate analysis.Section 1 introduces conventional operators for ‘and’ and ‘or’, and Section 2 provides suitable extensions and generalizations of them. Section 3 sets out the required least-squares techniques for fitting these generalized operators to data, first in the context of ANOVA problems and then in regression contexts, for single-connective (three-variable) models. The theoretical developments and examples from real data-sets demonstrate the utility of ‘and’ and ‘or’ as a means to cell-specific interpretations of interaction effects which can also readily be translated into English. Section 4 extends these developments to multivariate, multiple-connective models and discusses issues of generalizability. The paper concludes (Section 5) with a brief discussion of remaining unsolved problems, future prospects for more sophisticated models, and computer programs.     

Factors contributing to academic achievement: a Bayesian structure equation modelling study

In Iran, high school graduates enter university after taking a very difficult entrance exam called the Konkoor. Therefore, only the top-performing students are admitted by universities to continue their bachelor's education in statistics. Surprisingly, statistically, most of such students fall into the following categories: (1) do not succeed in their education despite their excellent performance on the Konkoor and in high school; (2) graduate with a grade point average (GPA) that is considerably lower than their high school GPA; (3) continue their master's education in majors other than statistics and (4) try to find jobs unrelated to statistics. This article employs the well-known and powerful statistical technique, the Bayesian structural equation modelling (SEM), to study the academic success of recent graduates who have studied statistics at Shahid Beheshti University in Iran. This research: (i) considered academic success as a latent variable, which was measured by GPA and other academic success (see below) of students in the target population; (ii) employed the Bayesian SEM, which works properly for small sample sizes and ordinal variables; (iii), which is taken from the literature, developed five main factors that affected academic success and (iv) considered several standard psychological tests and measured characteristics such as ‘self-esteem’ and ‘anxiety’. We then study the impact of such factors on the academic success of the target population. Six factors that positively impact student academic success were identified in the following order of relative impact (from greatest to least): ‘Teaching–Evaluation’, ‘Learner’, ‘Environment’, ‘Family’, ‘Curriculum’ and ‘Teaching Knowledge’. Particularly, influential variables within each factor have also been noted.     

Transportation policies for single and multi-objective transportation problem using fuzzy logic

In this paper, single and multi-objective transportation models are formulated with fuzzy relations under the fuzzy logic. In the single-objective model, objective is to minimize the transportation cost. In this case, the amount of quantities transported from an origin to a destination depends on the corresponding transportation cost and this relation is verbally expressed in an imprecise sense i.e., by the words ‘low’, ‘medium’, ‘high’. For the multi-objective model, objectives are minimization of (i) total transportation cost and (ii) total time for transportation required for the system. Here, also the transported quantity from a source to a destination is determined on the basis of minimum total transportation cost as well as minimum transportation time. These relations are imprecise and stated by verbal words such as ‘very high’, ‘high’, ‘medium’, ‘low’ and ‘very low’. Both single objective and multi-objective problems using Real coded Genetic Algorithms (GA and MOGA) are developed and used to solve the single level and bi-level logical relations respectively. The models are illustrated with numerical data and optimum results are presented.     

Writing practitioner case studies to help behavioural OR researchers ground their theories: application of the mangle perspective

Behavioural research into the practice of OR needs to be grounded. Case studies written by practitioners can potentially help address this need but currently most do not. The paper explores a way of describing OR projects that place the emphasis on the ‘actors’ who provide the motivating force and the consequences of their actions. The ‘mangle’ perspective focuses on the dynamic intertwining of people, technology and concepts; this can provide the basis for an insightful narrative describing the reality of the project in terms of the planned approach, the problems met and the outcomes. Two examples are given, one of a conventional model building exercise, the second of a ‘soft OR’ intervention: both describe projects conducted by practitioners for commercial purposes. It is concluded that, by using the mangle perspective, the OR case writer can winnow the wheat from the chaff in order to write a succinct informative narrative, a narrative that could be utilized by behavioural OR (BOR) researchers. It is further concluded that BOR researchers should engage with ‘practice theory’ to deepen their understanding of what actually happens in projects.     

Process and content: two ways of using SSM

Soft systems methodology (SSM) includes several ways of gaining a rich appreciation of the problem situation addressed. ‘Analysis One’, exploration of the intervention itself, is the subject here, since it is sparsely covered in the literature. The analysis is conducted in terms of three roles: ‘client’, ‘problem solver’ and ‘problem owner’. Whoever is in the role of ‘problem solver’ is free to define a list of possible ‘problem owners’, which brings many perspectives to bear on the situation. It was realized that ‘client’ and ‘problem solver’ should themselves feature in the ‘problem owner’ list. The ‘problem’ owned by the ‘problem solver’ is that of undertaking the intervention. This led to a realization that SSM is relevant to both the content of a perceived situation (SSMc) and the process of dealing with that content (SSMp). This development is described and illustrated by work in the National Health Service. The focus of the SSM use was to define the intellectual process for a service specification project which NHS professionals would themselves carry out.     

JOHN KNOPFMACHER, [ABSTRACT] ANALYTIC NUMBER THEORY,AND THE THEORY OF ARITHMETICAL FUNCTIONS

Abstract

Dedicated to the memory of John Knopfmacher (1937–1999)

In this paper some important contributions of John Knopfmacher to ‘Abstract Analytic Number Theory’ are described. This theory investigates semigroups with countably many generators (generalized ‘primes’), with a norm map (or a ‘degree map’), and satisfying certain conditions on the number of elements with norm less than x (Axiom A resp. Axiom A#), and ‘arithmetical’ functions defined on these semigroups.

It is tried to show some of the impact of John Knopfmachers ideas to the future development of number theory, in particular for the topics ‘arithmetical functions’ and ‘asymptotics in additive arithmetical semigroups’.     

Applying Stochastic Algorithms to a Locomotive Scheduling Problem

This paper addresses a problem common to all railway networks. Given a fixed train timetable and locomotives (or other forms of traction) of various types, each train must be allocated a locomotive. This paper examines the use of stochastic algorithms for such a problem. Two types of algorithm are used—a simple ‘local improvement’ method, performed successively from randomly chosen starting points, and a ‘simulated annealing’ approach. Both are found to give considerably better results than a deterministic method in current use, and the annealing approach is probably the better stochastic method.     

Improving productivity in finite element analysis through interactive processing

‘User friendly’ interactive pre- and post-processing programs allow major increases in productivity in finite element analysis. These programs are designed to reduce the amount of effort required to develop input and summarize output for large finite element programs. The gains in productivity, however, are not without cost. The use of this technology requires additional computer resources and user training. Although the increases in productivity are obvious to those who use this technology, management decisions are based on economic considerations. Therefore, engineering management must understand the benefits and the value of the increases in productivity to effectively evaluate the use of these tools. This paper discusses the concepts which form the basis for improving productivity through the use of interactive pre- and post-processing. Examples, taken from the authors' experience, are presented which illustrate the improvements in productivity which can be obtained using this technology.     

Evolutionary stability in extensive two-person games - correction and further development

The paper corrects a mistake concerning sufficient conditions for evolutionary stability in an earlier paper (MASS 5 (1983) 269–363). A new kind of decomposition of a symmetric extensive two-person game into a ‘top’ and an ‘abridgement’ is introduced. It is shown that regularity of a direct ESS imply a robustness property with respect to small payoff changes called essentiality. On the basis of the result on regularity and essentiality sufficient conditions for a regular limit ESS are obtained. These conditions make use of the decomposition into ‘top’ and ‘abridgement’. Finally it is proved that the sufficient conditions are satisfied for the many-period models with ritual fights and escalated conflicts introduced in the earlier paper.     

Can we have confidence in generic structures?

The system dynamics concept of ‘generic structure’ is dividable into three sub-types. This paper analyses the validity of these three, using both practical and theoretical perspectives. Firstly, a new set of measures is developed for generating validity—‘confidence’—amongst a group using generic structures in a practical modelling situation. It is concluded that different confidence criteria are implicitly employed; there is an argument for trading-off model precision and analytical quality for simplicity and ease of use and future research is needed to combine these ‘process’ and ‘content’ aspects of confidence. From a theoretical stance it is shown that with two of the sub-types a scientific notion of confidence is achievable whereas the third (‘archetypes’) involves merely metaphorical thinking. It is concluded that the theoretical status of archetypes requires further development, whilst ensuring that its benefits are retained.     

Developing a Visual Interactive Model for Corporate Cash Management

The operational research/management science journals contain an extensive literature that addresses the corporate cash management problem; yet few, if any, companies make use of any of this published work in their daily cash-management decision making. A review of the literature suggests that the reason for this lack of applications may well be poor problem formulation—the problems that are solved in the literature as ‘cash management’ problems evolve from a ‘hard systems’ view of real-world cash management. However, the problem as perceived by cash managers involves both dynamic and loosely structured components which are difficult to model using classical (i.e. ‘hard systems’) approaches.We therefore decided to approach the cash management problem as an experiment in the use of a novel visual interactive problem solving (VIPS) methodology. The aim of the experiment was to develop an implementable, visual interactive model to support daily cash management decision making. Working closely with a corporate cash manager, we first developed a visual model of his daily decision problem and then agreed on the feasible options and the interactive requirements. At this stage, the problem was sufficiently well defined for a mathematical model to be built and the visual model made ‘smart’.This paper discusses the results of this experiment and suggests that VIPS may have distinct advantages as a problem-solving technique in loosely structured, ‘messy’ problem situations.     

Evaluating models for classifying customers in retail banking collections

When seeking to establish a repayment strategy with delinquent borrowers, it is useful to determine how they are likely to behave, so that an optimal use of resources can be made. We examine two behavioural classifications (‘settle immediately’ versus ‘not settle immediately’, and ‘make some repayment’ versus ‘make no repayment’) and apply a variety of rules for predicting into which class each customer is likely to belong. Since no such rule will yield perfect predictions, the way in which performance is evaluated is crucial in choosing a good rule, and hence subsequently in obtaining accurate predictions of likely future behaviour. We examine some popular standard performance evaluation criteria, showing that they have major weaknesses. We describe and illustrate the use of an alternative measure that overcomes these weaknesses.     

Linear system solution by null‐space approximation and projection (SNAP)

Solutions of large sparse linear systems of equations are usually obtained iteratively by constructing a smaller dimensional subspace such as a Krylov subspace. The convergence of these methods is sometimes hampered by the presence of small eigenvalues, in which case, some form of deflation can help improve convergence. The method presented in this paper enables the solution to be approximated by focusing the attention directly on the ‘small’ eigenspace (‘singular vector’ space). It is based on embedding the solution of the linear system within the eigenvalue problem (singular value problem) in order to facilitate the direct use of methods such as implicitly restarted Arnoldi or Jacobi–Davidson for the linear system solution. The proposed method, called ‘solution by null‐space approximation and projection’ (SNAP), differs from other similar approaches in that it converts the non‐homogeneous system into a homogeneous one by constructing an annihilator of the right‐hand side. The solution then lies in the null space of the resulting matrix. We examine the construction of a sequence of approximate null spaces using a Jacobi–Davidson style singular value decomposition method, called restarted SNAP‐JD, from which an approximate solution can be obtained. Relevant theory is discussed and the method is illustrated by numerical examples where SNAP is compared with both GMRES and GMRES‐IR. Copyright © 2006 John Wiley & Sons, Ltd.