The state-of-art in mathematical creativity

Creativity is a topic which is often neglected within mathematics teaching. Usually teachers think that it is logic that is needed in mathematics in the first place, and that creativity is not important and learning mathematics. On the other hand, if we consider a mathematician who develops new results in mathematics. we cannot overlook his/her use of the creative potential. Thus, the main questions are as follows: What methods could be used to foster mathematical creativity within school situations? What scientific knowledge, i.e. research results, do we have on the meaning of mathematical creativity?     

Recognising mathematical creativity in schoolchildren

Examples of tasks designed to recognise creative thinking within mathematics, used with 11–12-year-old pupuls, are described. The first construct empoyed in the design of these tasks is the ability to overcome fixation. Sometimes pupils demonstrate content-universe fixation, by restricting their thinking about a problem to an insufficient or inappropriate range of elements. Other times they show algorithmic fixation by continuing to adhere to a routine procedure or stereotype response even when this becomes inefficient or inappropriate. The second construct employed is that of divergent production, indicated by flexibility and originality in mathematical tasks to which a large number of appropriate responses are possible. Examples of three categories of such tasks are described: (1) problem-solving, (2) problem-posing, and (3) redefinition. Examples of pupils’ responses to various tasks are used to argue that they do indeed reveal thinking that can justifiably be described as creative. The relationship to conventional mathematics attainment is discussed-mathematics attainment is seen to limit but not to determine mathematical creativity.     

The methods of fostering creativity through mathematical problem solving

Which methods could be used to foster mathematical creativity in school situations? The following topics are treated with the respect to this question: 1. “Open-ended approach” and “From problem to problem”, 2. Relation to mathematical creativity, 3. Teacher’s belief and the mathematics textbook.     

Fostering creativity through instruction rich in mathematical problem solving and problem posing

Although creativity is often viewed as being associated with the notions of “genius” or exceptional ability, it can be productive for mathematics educators to view creativity instead as an orientation or disposition toward mathematical activity that can be fostered broadly in the general school population. In this article, it is argued that inquiry-oriented mathematics instruction which includes problem-solving and problem-posing tasks and activities can assist students to develop more creative approaches to mathematics. Through the use of such tasks and activities, teachers can increase their students’ capacity with respect to the core dimensions of creativity, namely, fluency, flexibility, and novelty. Because the instructional techniques discussed in this article have been used successfully with students all over the world, there is little reason to believe that creativity-enriched mathematics instruction cannot be used with a broad range of students in order to increase their representational and strategic fluency and flexibility, and their appreciation for novel problems, solution methods, or solutions.     

Connecting Lisp-Stat to COM

Abstract

This article describes an interface between Lisp-Stat and Microsoft's Component Object Model (COM). This interface can be used, for example, to retrieve data stored in a spreadsheet or a database. After providing some brief background on Lisp-Stat and COM, the interface is presented through a series of examples. Issues in implementation and integration relevant to other statistical systems and other component systems are discussed.     

Comparisons of mathematical creativity,some personality and biographical factors of middle school dropouts and stayins

Middle school dropouts and stayins were compared on mathematical creativity, some personality and biographical factors. Verbal and non‐verbal mathematical creativity tests, a Hindi adaptation of the Thorndike dimensions of temperament test and a biographical inventory were used on 70 dropouts and 100 stayins male students, aged 11+ to 13+ years, randomly selected, from Sultanpur District, India. The results showed that: (1) mathematical creativity of dropouts was found to be lower than stayins; (2) dropouts were found to be sociable, accepting, reflective, lethargic and casual in nature whereas stayins were found to be solitary, critical, practical, premeditated, active and responsible in nature; and (3) the level of family income, professional background of the family, parents’ education, standard of living, interest‐patterns, attitude and level of aspiration of the stayins were found to be higher than of the dropouts.     

Some applications of mathematical programming to problems in mathematical physics

Siberian Mathematical Journal -     

Introducing mathematical modelling to undergraduates

Some motivation for, and thoughts behind, the development of a course in mathematical modelling are described. The structure of the course and some of the problems of teaching and assessment are reviewed. The course has the characteristics that it is introductory and discipline‐independent.     

The ability of students and teachers to use counter-examples to justify mathematical propositions: a pilot study in South Korea and Hong Kong

Counter-examples, which are a distinct kind of example, have a functional role in inducing logically deductive reasoning skills in the learning process. In this investigation, we compare the ability of students and prospective teachers in South Korea and Hong Kong to use counter-examples to justify mathematical propositions. The results highlight that South Korean students performed better than Hong Kong students at justifying propositions using counter-examples in algebra problems, but both did equally well in geometry problems. In terms of the prospective teachers’ ability to justify propositions using counter-examples in two particular topics, properties of the absolute value function and parallelogram, Hong Kong prospective teachers performed relatively weakly in the absolute value problem but better in the parallelogram problem compared with South Korean prospective teachers. The weaknesses and strengths of students and prospective teachers in generating counter-examples associated with logical reasoning in mathematical contexts in the two regions indicate ways of improving the effectiveness of learning and teaching mathematics through the use of counter-examples.     

According to Davis: Connecting principles and practices

In this article, the author allows Robert B. Davis to state for himself his own Principles concerning how children learn, and how teachers can best teach them. These principles are put forward in Davis’ own words along with detailed documentation. The author goes on compare Davis’ words with his practices. A single Davis video (Towers of Hanoi) is analyzed to determine if, and to what extent, his principles are evident in his teaching of this lesson.     

Tests of creativity in mathematics

A considerable amount of research on ways of testing creative ability in mathematics is now being written and this takes two forms. On the one hand there are fundamental researches which try to provide diagnostic test items on mathematical creativity and then to use these to find the relationship between this variable and a number of others. On the other hand there are attempts to create assessment items which try to measure the end‐product of a modern discovery‐based mathematics curriculum. This paper examines the criteria and form of some of these items.     

A hybrid approach to discrete mathematical programming

The dynamic programming and branch-and-bound approaches are combined to produce a hybrid algorithm for separable discrete mathematical programs. Linear programming is used in a novel way to compute bounds. Every simplex pivot permits a bounding test to be made on every active node in the search tree. Computational experience is reported.     

An introduction to mathematical physiology and biology

Book Review

An introduction to mathematical physiology and biologyJ. Mazumdar: Cambridge University Press, 1989, xi + 208pp., £27.50 (hardback), £9.95 (paperback)     

A mathematical approach to HIV infection dynamics

In order to obtain a comprehensive form of mathematical models describing nonlinear phenomena such as HIV infection process and AIDS disease progression, it is efficient to introduce a general class of time-dependent evolution equations in such a way that the associated nonlinear operator is decomposed into the sum of a differential operator and a perturbation which is nonlinear in general and also satisfies no global continuity condition. An attempt is then made to combine the implicit approach (usually adapted for convective diffusion operators) and explicit approach (more suited to treat continuous-type operators representing various physiological interactions), resulting in a semi-implicit product formula. Decomposing the operators in this way and considering their individual properties, it is seen that approximation–solvability of the original model is verified under suitable conditions. Once appropriate terms are formulated to describe treatment by antiretroviral therapy, the time-dependence of the reaction terms appears, and such product formula is useful for generating approximate numerical solutions to the governing equations. With this knowledge, a continuous model for HIV disease progression is formulated and physiological interpretations are provided. The abstract theory is then applied to show existence of unique solutions to the continuous model describing the behavior of the HIV virus in the human body and its reaction to treatment by antiretroviral therapy. The product formula suggests appropriate discrete models describing the dynamics of host pathogen interactions with HIV1 and is applied to perform numerical simulations based on the model of the HIV infection process and disease progression. Finally, the results of our numerical simulations are visualized and it is observed that our results agree with medical and physiological aspects.     

Rival approaches to mathematical modelling in immunology

In order to formulate quantitatively correct mathematical models of the immune system, one requires an understanding of immune processes and familiarity with a range of mathematical techniques. Selection of an appropriate model requires a number of decisions to be made, including a choice of the modelling objectives, strategies and techniques and the types of model considered as candidate models. The authors adopt a multidisciplinary perspective.