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Difficulties in understanding mathematics: an approach related to working memory and field dependency

Onwumere, Onyebuchi (2009) Difficulties in understanding mathematics: an approach related to working memory and field dependency. PhD thesis, University of Glasgow.

It is commonly agreed that learning with understanding is more desirable than learning by rote. Understanding is described in terms of the way information is represented and structured in the memory. A mathematical idea or procedure or fact is understood if it is a part of an internal network, and the degree of understanding is determined by the number and the strength of the connections between ideas. When a student learns a piece of mathematical knowledge without making connections with items in his or her existing networks of internal knowledge, he or she is learning without understanding.

Learning with understanding has progressively been elevated to one of the most important goals for all learners in all subjects. However, the realisation of this goal has been problematic, especially in the domain of mathematics where there are marked difficulties in learning and understanding. The experience of working with learners who do not do well in mathematics suggests that much of the problem is that learners are required to spend so much time in mathematics lessons engaged in tasks which seek to give them competence in mathematical procedures. This leaves inadequate time for gaining understanding or seeking how the procedures can be applied in life.

Much of the satisfaction inherent in learning is that of understanding: making connections, relating the symbols of mathematics to real situations, seeing how things fit together, and articulating the patterns and relationships which are fundamental to our number system and number operations. Other factors include attitudes towards mathematics, working memory capacity, extent of field dependency, curriculum approaches, the classroom climate and assessment. In this study, attitudes, working memory capacity and extent of field dependency will be considered. The work will be underpinned by an information processing model for learning.

A mathematics curriculum framework released by the US National Council of Teachers of Mathematics (NCTM, 2000) offers a research-based description of what is involved for students to learn mathematics with understanding. The approach is based on “how learners learn, not on “how to teach”, and it should enable mathematics teachers to see mathematics from the standpoint of the learner as he progresses through the various stages of cognitive development.

The focus in the present study is to try to find out what aspects of the process of teaching and learning seem to be important in enabling students to grow, develop and achieve. The attention here is on the learner and the nature of the learning process.

What is known about learning and memory is reviewed while the literature on specific areas of difficulty in learning mathematics is summarised. Some likely explanations for these difficulties are discussed. Attitudes and how they are measured are then discussed and there is a brief section of learner characteristics, with special emphasis on field dependency as this characteristic seems to be of importance in learning mathematics. The study is set in schools in Nigeria and England but the aim is not to make comparisons.

Several types of measurement are made with students: working memory capacity and extent of field dependency are measured using well-established tests (digit span backward test and the hidden figure test). Performance in mathematics is obtained from tests and examinations used in the various schools, standardised as appropriate. Surveys and interviews are also used to probe perceptions, attitudes and aspects of difficulties. Throughout, large samples were employed in the data collection with the overall aim of obtaining a clear picture about the nature and the influence of attitudes, working memory capacity and extent of field dependency in relation to learning, and to see how this was related to mathematics achievement as measured by formal examination.

The study starts by focussing on gaining an overview of the nature of the problems and relating these to student perception and attitudes as well as working memory capacity. At that stage, the focus moves more towards extent of field dependency, seen as one way by which the fixed and limited working memory capacity can be used more efficiently. Data analysis was in form of comparison and correlation although there are also much descriptive data.

Some very clear patterns and trends were observable. Students are consistently positive towards the more cognitive elements of attitude to mathematics (mathematics is important; lessons are essential). However, they are more negative towards the more affective elements like enjoyment, satisfaction and interest. Thus, they are very realistic about the value of mathematics but find their experiences of learning it much more daunting. Attitudes towards the learning of mathematics change with age. As students grow older, the belief that mathematics is interesting and relevant to them is weakened, although many still think positively about the importance of mathematics.

Loss of interest in mathematics may well be related to an inability to grasp what is required and the oft-stated problem that it is difficult trying to take in too much information and selecting what is important. These and other features probably relate to working memory overload, with field dependency skills area being important. The study identified clearly the topics which were perceived as most difficult at various ages. These topics involved ideas and concepts where many things had to be handled cognitively at the same time, thus placing high demands on the limited working memory capacity.

As expected, working memory capacity and mathematics achievement relate strongly while extent of field dependency also relates strongly to performance. Performance in mathematics is best for those who are more field-independent. It was found that extent of field dependency grew with age. Thus, as students grow older (at least between 12 and about 17), they tend to become more field-independent. It was also found that girls tend to be more field-independent than boys, perhaps reflecting maturity or their greater commitment and attention to details to undertake their work with care during the years of adolescence.

The outcomes of the findings are interpreted in terms of an information processing model. It is argued that curriculum design, teaching approaches and assessment which are consistent with the known limitations of the working memory must be considered during the learning process. There is also discussion of the importance of learning for understanding and the problem of seeking to achieve this while gaining mastery in procedural skills in the light of limited working memory capacity. It is also argued that positive attitudes towards the learning in mathematics must not only be related to the problem of limited working memory capacity but also to ways to develop increased field independence as well as seeing mathematics as a subject to be understood and capable of being applied usefully.

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Mathematics > Analysis of PDEs

Title: elliptic operators and boundary problems in spaces of generalized smoothness.

Abstract: The paper contains a survey of the results obtained during the last ten years in the theory of elliptic boundary problems in Hörmander function spaces, developed by the authors, and other related results of modern analysis. The basics of this theory and some of its applications are systematically expounded in the monograph "Hörmander Spaces, Interpolation, and Elliptic Problems" (De Gruyter, Berlin/Boston, 2014) by the first two authors of the survey.

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STUDENTS’ DIFFICULTIES IN COMPREHENDING MATHEMATICAL WORD PROBLEMS IN ENGLISH LANGUAGE LEARNING CONTEXTS

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