An investigation into the use of microcomputers to teach mathematical problem‐solving skills to 13 year olds

This is an account of an experiment to teach mathematical problem‐solving skills to 13‐year‐old students with the aid of microcomputers. Twenty‐one children worked through six microcomputer program‐generated problem situations in two and a half hours. Nine of the children worked ‘hands on’ with microcomputers. Twelve of the children were class taught, with a single microcomputer used for demonstration purposes. A comparison of pre‐test and post‐test scores on tests indicated that no significant gains in achievement were made by either group. Informal observations suggest a positive affective response. The findings are discussed at length and possible explanations for the lack of achievement are conjectured.     

How does imposing a step-by-step solution method impact students’ approach to mathematical word problem solving?

This paper presents the second phase of a larger research program with the purpose of exploring the possible consequences of a gap between what is done in the classroom regarding mathematical word problem solving and what research shows to be effective in this particular field of study. Data from the first phase of our study on teachers’ self-proclaimed practices showed that one-third of elementary teachers from the region of Quebec require their students to follow a specific sequential problem-solving method, known as the ‘what I know, what I look for’ method. These results led us to hypothesize that the observed gap may have an impact on students’ comprehension of mathematical word problems. The use of this particular method was the foundation for us to study, in the second phase, the effect of the imposition of this sequential method on students’ literal and inferential understanding of word problems. A total of 278 fourth graders (9–10 years old) solved mathematical word problems followed by a test to assess their understanding of the word problems they had just solved. The results suggest that the use of this problem solving method does not seem to improve or impair students’ understanding. From a more fundamental point of view, our study led us to the conclusion that the way word problem solving is addressed in the mathematics classroom, through sequential and inflexible methods, does not help students develop their word problem solving competence.     

What do the arithmetic,geometric and harmonic means tell us in length-biased sampling?

Properties of the A.M., G.M. and H.M. for length-biased distributions are studied in a nonparametric fashion. Unbiased estimation of the coefficient of variation is considered, and a characterization of length-biased distributions is also made in this light.     

Fifth-grade students�� approaches to and beliefs of mathematics word problem solving: a large sample Hungarian study

Mathematical word problems used in Verschaffel et al.??s (Learning and Instruction 7:339?C359, 1994) study were applied in several follow-up studies. The goal of the present study was to replicate and extend the results of this line of research in a large sample of Hungarian students using an alternative set of data-gathering and data-analysis techniques. 4,037 students forming a nationwide representative sample of the Hungarian fifth-grade student population (aged 10?C11) completed the test. The test contained five word problems from the list of 10 P(??problematic)-items from Verschaffel et al.??s test. In contrast to all previous research in this domain, we used a multiple-choice format, where three options were given for each task: (a) routine-based, non-realistic answer, (b) numerical response that does take into account realistic considerations, (c) a realistic solution stating that the task cannot be solved. The hypotheses of this study were: (1) Students?? responses will confirm previous results, i.e. upper elementary school students prefer to respond to P-items by means of the routine-based answer; (2) Most students will demonstrate a more or less consistent preference for a given answer type (a, b or c) over problems; (3) Students?? school math marks will have low correlation indices with students?? achievement on these word problems. Our results confirm student??s overall tendency to follow non-realistic approaches when doing school word problem solving. The tendency even holds when confronting students with various kinds of realistic answers. Our results show that students demonstrate response patterns over problems, and that the correlation with math school performance is significant but small.     

A modification of the method of linearization for solving a mathematical programming problem on a simple set of “parallelepiped” type

We propose a modification of B. N. Pshenichnyi's method of linearization for solving mathematical programming problems on a simple set of parallelepiped type. We study in detail the computational aspects and the efficiency of the approach in question. Two tables. Bibliography: 3 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 24–28.     

How can self-regulated learning support the problem solving of third-grade students with mathematics anxiety?

The study compares 140 third-grade Israeli students (lower and higher achievers) who were either exposed to self-regulated learning (SRL) supported by metacognitive questioning (the MS group) or received no direct SRL support (the N_MS group). We investigated: (a) mathematical problem solving performance; (b) metacognitive strategy use in three phases of the problem-solving process; and (c) mathematics anxiety. Findings indicated that the MS students showed greater gains in mathematical problem solving performance than the N_MS students. They reported using metacognitive strategies more often, and showed a greater reduction in anxiety. In particular, the lower MS achievers showed these gains in the basic and complex tasks, in strategy use during the on-action phase of the problem solving process and a decrease in negative thoughts. The higher achievers showed greater improvement in transfer tasks and an increase in positive thoughts towards mathematics. Both the theoretical and practical implications of this study are discussed.     

What portion of the sample makes a partial sum asymptotically stable or normal?

Summary Let a sequence of independent and identically distributed random variables with the common distribution function in the domain of attraction of a stable law of index 0<2 be given. We show that if at each stage n a number k _n depending on n of the lower and upper order statistics are removed from the n-th partial sum of the given random variables then under appropriate conditions on k _n the remaining sum can be normalized to converge in distribution to a standard normal random variable. A further analysis is given to show which ranges of the order statistics contribute to asymptotic stable law behaviour and which to normal behaviour. Our main tool is a new Brownian bridge approximation to the uniform empirical process in weighted supremum norms.Work done while visiting the Bolyai Institute, Szeged University, partially supported by a University of Delaware Research Foundation Grant     

A note on the principle of mathematical induction,intuition and consciousness in the light of ideas of Poincaré and Galois

In his argumentation for the non-computability of thought-processes (in general) Penrose is invoking Gödel’s theorem (see [R. Penrose, Shadows of the Mind – A search for the Missing Science of Consciousness, Oxford University Press, Oxford, 1994]). It is the aim with the following note to indicate that the same effect may be obtained in a simpler and possibly also more fundamental way. This does not necessarily mean that I fully believe in Penrose’s thesis – the question is still largely open – but I think that my note indicates that there are a lot of items that remains to be clarified before a satisfactory scientific consensus will be reached. There is a huge gap between the precision of strict scientific contexts and those where this kind of processes are going on. At the same time we will see that the same kind of ideas had been impinging themselves on mathematicians like Poincaré and Galois, like Penrose himself of a very intuitive kind. It is plausible that the solution of the enigma of the scientific character of processes referring back to themselves lies in deep properties of autonomous systems. The self-referential character of the interpretations in Gödel’s theorem is quite central. This will be the subject of a forthcoming paper.     

On a modification of a problem of Bialostocki, Erd?s, and Lefmann

For positive integers m and r, one can easily show there exist integers N such that for every map Δ:{1,2,…,N}→{1,2,…,r} there exist 2m integers

x₁<?<x_m<y₁<?<y_m,     

What is so special about mathematical texts? Analyses of common claims in research literature and of properties of textbooks

This study surveys claims in research articles regarding linguistic properties of mathematical texts, focusing on claims supported by empirical or logical arguments. It also performs a linguistic analysis to determine whether some of these claims are valid for school textbooks in mathematics and history. The result of the survey shows many and varying claims that mainly describe mathematical texts as highly compact, precise, complex, and containing technical vocabulary. However, very few studies present empirical support for their claims, and the few empirical studies that do exist contradict the most common, and unsupported, claims, since no empirical study has shown mathematical texts to be more complex than texts from other subjects, and any significant differences rather indicate the opposite. The linguistic analysis in this study is in line with previous empirical studies and stands in contrast to the more common opinion in the unsupported claims. For example, the mathematics textbooks have significantly shorter sentences than the history textbooks.     

On the Fu?ik spectrum of the wave operator and an asymptotically linear problem

We study generalized solutions of the nonlinear wave equation

utt−uss=au⁺−bu⁻+p(s,t,u),     

Michèle Audin,Fatou, Julia,Montel: the Great Prize of mathematical sciences of 1918, and beyond

Wills have long been a staple of the family historian but their study in the context of the history of mathematics seems to have received little attention. Yet the historian of mathematics or of family seeks answers to the same questions: what can we learn about the testator's family, his friends, his networks and interests, and the milieu in which he lived and worked? This article presents the will of Henry Briggs. It sheds an interesting light on college life in seventeenth-century Oxford and shows how wide-ranging was Briggs's circle of friends and relatives.     

Exploring the relationship between metacognitive and collaborative talk during group mathematical problem-solving – what do we mean by collaborative metacognition?

The purpose of this study was to enhance our understanding of the relationship between collaborative talk and metacognitive talk during group mathematical problem-solving. Research suggests that collaborative talk may mediate the use of metacognitive talk, which in turn is associated with improved learning outcomes. However, our understanding of the role of group work on the individual use of metacognition during problem-solving has been limited because research has focused on either the individual or the group as a collective. Here, primary students (aged nine to 10) were video-recorded in a naturalistic classroom setting during group mathematical problem-solving sessions. Student talk was coded for metacognitive, cognitive and social content, and also for collaborative content. Compared with cognitive talk, we found that metacognitive talk was more likely to meet the criteria to be considered collaborative, with a higher probability of being both preceded by and followed by collaborative talk. Our results suggest that collaborative metacognition arises from combined individual and group processes.     

What part of the concept of acceleration is difficult to understand: the mathematics, the physics, or both?

In this paper, details of student difficulties in understanding the concept of acceleration and the mathematical and physical/intuitive sources of these are delineated by utilizing the teaching experiment methodology. As a result of the study, two anchoring analogies are proposed that can be used as a diagnostic tool for students’ alternative conceptions. These can be used in teaching to highlight the peculiarity of acceleration concept. This study portrays how seeing acceleration as ‘rate of change’ of a quantity (velocity) and recognizing the consequences of such a definition are hindered in certain ways which in turn negatively affect learning the concept of force. This is also an example that illustrates that a rather “simple” mathematical concept (i.e., rate of change) for the expert can become a complex phenomenon when embedded in a physical concept (i.e., acceleration) which is consistently found to be as a misconception among learners at various levels that is widely occurring and very resistant to change.     

Fixed Charge Transportation Problems: a new heuristic approach based on Lagrangean relaxation and?the?solving of core problems

In this paper the Fixed Charge Transportation Problem is considered. A new heuristic approach is proposed, based on the intensive use of Lagrangean relaxation techniques. The more novel aspects of this approach are new Lagrangean relaxation and decomposition methods, the consideration of several core problems, defined from the previously computed Lagrangean reduced costs, the heuristic selection of the most promising core problem and the final resort to enumeration by applying a branch and cut algorithm to the selected core problem. For problems with a small ratio of the average fixed cost to the average variable cost (lower than or equal to 25), the proposed method can obtain similar or better solutions than the state-of-art algorithms, such as the tabu search procedure and the parametric ghost image processes. For larger ratios (between 50 and 180), the quality of the obtained solutions could be considered to be halfway between both methods.     

What is the role and contribution of models to management and research in the health services? A view from Europe

Modelling in different forms has long been regarded as a cornerstone of health OR as in other fields of application. Models are now tending to become a standard tool in health services management and research. What are the lessons we as operational researchers have learnt during this development? How can health care managers and health service researchers benefit from modelling — with or without OR-analysts to guide them between the pitfalls? After an introductory overview concerning the nature and objectives of modelling, examples will be given of modelling applications from different health service areas in order to illustrate the versatility of the method. Further, the choice of methods and models is discussed with special attention to the problems of interpretation and implementation of results. From this overview some conclusions are drawn with regards to advantages and disadvantages of modelling as tools for policy planning and decision-making in the health area. Finally some observations are made concercing the conditions for the future development in the field.     

Carpenter,tractors and microbes for the development of logical–mathematical thinking – the way 10th graders and pre-service teachers solve thinking challenges

The objective of this case study was to investigate the ability of 10th graders and pre-service teachers to solve logical–mathematical thinking challenges. The challenges do not require mathematical knowledge beyond that of primary school but rather an informed use of the problem representation. The percentage of correct answers given by the 10th graders was higher than that of the pre-service teachers. Unlike the 10th graders, some of whom used various strategies for representing the problem, most of the pre-service teachers’ answers were based on a technical algorithm, without using control processes. The obvious conclusion drawn from the findings supports and recommends expanding and enhancing the development of logical–mathematical thinking, both in specific lessons and as an integral part of other lessons in pre-service frameworks.     

On the Brezis–Nirenberg problem on S3, and a conjecture of Bandle–Benguria

We consider the following Brezis–Nirenberg problem on ${\mathbf{S}}^3$

$?{\mathrm{\Delta }}_{{\mathbf{S}}^3}u=\lambda u+u^5\text{in}D,u>0\text{in}D\text{and}u=0\text{on }?D,$

where D is a geodesic ball on ${\mathbf{S}}^3$ with geodesic radius ${\theta }_1$, and ${\mathrm{\Delta }}_{{\mathbf{S}}^3}$ is the Laplace–Beltrami operator on ${\mathbf{S}}^3$. We prove that for any $\lambda <?\frac{3}{4}$ and for every ${\theta }_1<\pi$ with $\pi ?{\theta }_1$ sufficiently small (depending on λ), there exists bubbling solution to the above problem. This solves a conjecture raised by Bandle and Benguria [J. Differential Equations 178 (2002) 264–279] and Brezis and Peletier [C. R. Acad. Sci. Paris, Ser. I 339 (2004) 291–394]. To cite this article: W. Chen, J. Wei, C. R. Acad. Sci. Paris, Ser. I 341 (2005).