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The research reported in this thesis is divided up into three main sections. The first section begins with a brief review of the literature on methods of conducting psychological research in the classroom. It is argued that an interest in the psychology of education must be followed by a methodology which takes into account the fact that schools are social institutions within our society, and that they have functions other than the teaching of cognitive skills. Learning which takes place in schools may therefore be seen as being embedded within a social milieu. The classroom research therefore begins with an extended period of observations and interviews. Initially, the investigator operates as a non-participant, and gradually, as his presence becomes an unnoticed part of the school's routine, certain limited interactions are carried out with the children. Conversations and informal interviews are also held with the head and members of the teaching staff. From all of these encounters, a simple model of mathematical learning in the classroom is put forward. This is supported by two scenarios taken from actual classroom situations. A hypothesis on mathematical learning in the classroom is then developed which states that children must experience a certain amount of autonomous activity if they are to formulate 'higher order' strategies for dealing with mathematical representations, structures, and problems. Section two of the thesis contains a series of experiments which are designed to test aspects of this hypothesis in a controlled setting. This is accomplished by using three specially designed portable 'toys' which each contain rules or patterns relating to mathematical systems. Positive evidence is found for the claim that an autonomous condition is more effective than a yoked, control or combined experience (hybrid) in promoting mathematical conceptualisation as scored by written and verbal measures. In the third section of the thesis, the toys are brought into the classrooms in an attempt to integrate them with more formal instruction. Results of these efforts are assessed and documented via interviews, photographs, and samples of classroom work. Educational implications and suggestions for further research are then briefly discussed

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There is at present a national concern about basic mathematics, particularly at school leavers' level. In this connection, this thesis is an attempt to cope with the problem of the mathematical competence of engineering students at the interface between school and university. Engineering students were given individual diagnosis by means of a test and tutorial discussion and by means of an approach through thinking aloud combined with observation and retrospective interview. A common characteristic of the two approaches to individual diagnosis is that they require the presence of a tutor for the diagnosis. In the belief that students can diagnose themselves, if they are provided with the appropriate tools, a battery of mathematics diagnostic tests was developed. The items in this battery were based on the topics and level of difficulty mentioned by lecturers in a survey carried out with the purpose of finding out the mathematics needed in first year engineering courses at university, particularly during the first term. As the battery of tests was being developed, the mathematical needs of sixth form school leavers and of newly enrolled engineering students were identified. It was found that both groups of students had difficulty with those topics which involved the use of formulae, and that the knowledge of concepts and numerical values tended to be forgotten more rapidly than other knowledge. Following this diagnosis, materials in algebra and trigonometry were developed for revision purposes, based on students' actual needs. These employed the principles of programmed learning, which were found to be useful and to have certain advantages over traditional textbooks.

It is known that motivating people to learn mathematics can be difficult. My research explores adults’ motivation to learn mathematics and focuses on learners who are overcoming many barriers to study in the workplace, in classes organised and funded by their trade union. The adults are aiming to gain a formal qualification, using less formal learning approaches, in a non-traditional context, hence this research offers teachers in more conventional education settings an opportunity to learn about successful alternative practice. Using a grounded theory approach on qualitative data and by exploring the findings within a range of sources of research literature into motivation and mathematics learning for adults, the research distinguishes between initial motivation to re-engage with learning and motivation to continue learning. It identifies the dynamic interplay in motivation between the personal needs or goals of individual learners, and the influence of other members of faceto- face learning groups. This interplay is seen as shaped in the wider context of UK society, and, in particular, by the role of trade unions. Adults in trade union organised classes report being able to successfully develop their mathematical skills and confidence through the use of ‘collective’ learning approaches, which develop positive social and emotional encounters in the classroom that are different from their previous experiences. Developing confidence while learning mathematics helps to shape their identities and, for this sample, has considerable influence on their motivations both inside and outside the classroom. The significance of developing confidence is reinforced by the adults’ use of emotional language when reporting changes in their feelings towards mathematics from negative to positive and their motivation to learn, described as an adult’s ‘Affective Mathematical Journey’. This emphasises the importance of considering the emotional connection to motivation and cognitive development when reflecting on professional practice associated with adults learning mathematics.

This thesis comprises two main studies which sort to investigate the effect that the study of advanced mathematics had on performance on spatial tasks. The first cross-sectional study tested both pre- and post-advanced study students and found an advantage for the mathematicians in a general spatial ability, but no clear evidence of an education level/group interaction. The second longitudinal study tested students at two time points, before and after a year of advanced study. Again, the mathematicians showed higher spatial skills at both time points, but there was no interaction between time and group. Bayesian analyses of the data revealed moderate to strong evidence for the null hypothesis: that there was no formal discipline value of studying advanced mathematics in terms of an effect on spatial skills.

"Mathematics is difficult and especially so for girls". The researcher had heard this but believed that it was success in examinations that proved more difficult for students rather than mathematics itself and that often girls underestimated their ability to achieve in this curriculum area. This is the case study focussing on four secondary schools, with varying percentages for girls, in rural Kenya. Kenya, in sub-Saharan Africa, is a country where there is no universal primary education (UPE) and education is not compulsory. Whilst looking at attitudes of all students, the emphasis has tended to be on the girls. The researcher believes that the evidence from these schools in sufficient to suggest that Kenyan schools should not necessarily accept the female prejudices against mathematics that research has found in first world countries. These findings both confirmed and contradicted areas of previous research on gender and mathematics. Whilst the majority of mathematics' teachers were male, the results endorse the literature which shows that this does not disadvantage the female students. However, expecting to find the girls marginalised in a country where a pedagogy of difference exists, in a subject considered a male domain, the researcher found that this was not the case: these findings tended to corroborate those of Driver (1980), Parry (1997) and Mittelberg and Lev-Ari (1999) in that female students, when motivated, are able to cope with mathematics. Many of the girls were enthusiastic about mathematics when they entered the secondary school and they were aware of the importance of mathematics for their future careers. The researcher was not able to ascertain that girls would perform better in a single-sex environment. The researcher believes that it is the expectation of a high level of mathematics for all students, together with the intensive syllabus that causes most Kenyan students to perform very badly in the mathematics' examination at the end of secondary schooling.

The analysis of some of the problems associated with teaching mathematics to science students provided in this thesis is a philosophical one, and as such claims to clarify rather than solve these problems. An important aspect of the thesis is the criticisms it offers of the philosophical and methodological assumptions underlying research within educational technology. In particular, the attempt to state the end point of educational processes in terms of precise statements of behaviour has been criticised, not by suggesting that there are other, non-observable (i. e. mental) outcomes, but by demonstrating that the intended behaviours cannot be stated in advance. This criticism has been based on Wittgenstein's analysis of understanding, which Wittgenstein calls a 'grammatical' analysis since, crudely, it consists of looking at how words are used. The purpose of the analysis, in this thesis, has been to show that educational technologists, in looking for behaviour corresponding to understanding, have misunderstood the use of the word 'understanding'. The context for the discussion of understanding has been provided by Kuhn's view of the nature of science and the analysis has been linked With Wittgenstein's criticism of the traditional notion of a concept since this has a bearing on assumptions made about what is learnt in the mathematics class and taken over into the science class. This analysis if accepted, demonstrates that in order to study the problems associated with mathematics for science courses, one cannot specify, in behavioural terms, what the student must transfer. Instead, and this is the main conclusion of the thesis, one must look at the criteria for understanding used in the two subjects.

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