Proceedings of the Day Conference held at the King's College London, February 2004.
Calculus is highly symbolic in nature and therefore students often try to get through calculus by manipulating the symbols without understanding the meaning of such symbols (i.e. having a procedural but not a conceptual understanding of the topics in calculus). Educators are looking for ways to help students achieve higher levels of conceptual understanding. This study explored Science Foundation Year students' graphical, numerical and algebraic understanding of the derivative concepts after differential calculus course. The course was designed to develop students' conceptual understanding of the derivative concept.
We engaged a smallish sample of students in a designed situation based on equivalence relations (from an expert point of view). The students were different from each other in age and educational background, and all were unfamiliar with the formal treatment of equivalence relations. The study was conducted by holding individual in-depth task-based interviews, in which we aimed at investigating the ways that students organize the given situation, rather than teaching them any particular ways of organizing that. As result, I will report a certain way of organizing the given situation, from that a 'new' definition of equivalence relations, and consequently a new representation for them, is emerged; a definition that seems to be overlooked by the experts.
Young bilingual students in the UK face the challenge of learning mathematics and learning English simultaneously. In this paper, I draw on work in bilingual education concerning the role of participation in meaningful interaction in language acquisition. Using an approach to analysis based on ideas in discursive psychology, I present an analysis of a short extract of interaction between a Year 1 learner of English as an additional language (EAL) and his teacher in a mathematics lesson. The student appears to make 'guesses' in response to the teacher's questions. My analysis suggests, however, that this behaviour arises from the socially organized structure of the interaction, as much as from the student's arithmetic proficiency.
Two year 7 classes in a Manchester school were taught multiplication, division and fractions. An experimental group was taught these numerical skills, but their teaching program included practical problem solving, based upon activity theory principles, as an integral component. A control group practised their number skills in more traditional abstract contexts. As expected, the control group was not able to transfer number fluency to practical problem solving tasks. The experimental group, however, demonstrated a problem solving ability at higher GCSE level and achieved a significant improvement in mean scores over the dynamic assessment that followed the teaching program. The dynamic core of this assessment was computer based and there was a strong negative relationship between hints given by the computer and residual gains. Analyses of the computer records have provided important clues to guide a qualitative analysis of video records of the teaching program.
Tonbridge Grammar School, Tonbridge, Kent.
An outline of a development project initiated to prevent the continuation of spoon feeding teaching at a grammar school. The report is covers the background information, research process, some examples of student work, and finally gives some tentative conclusions.
Teacher change in mathematics education is recognised to be a difficult and at times painful process. This is particularly so for generalist primary teachers, who have often had negative experiences of mathematics. In this paper I explore how one teacher developed a desire to be a mathematics teacher, thus enabling her to engage with change despite its difficulty. Drawing on theories of identity and situated learning, I conceive of motivation in terms of desire and argue that emotion is a potentially powerful element of mathematics teacher education.
We found, in two separate studies (1996 and 2002), that high attaining Year 10 students in English schools tend to produce empirical proofs, though many of them seem able to appreciate some of the qualities of more powerful proofs. Students rate algebraic proofs highly, often for superficial reasons, though we found that in the second, longitudinal, study they were more discriminating in Year 10 than they had been in Year 9.
In recent years there have been several films featuring a mathematician as the central character. In this article I focus on four of these: A Beautiful Mind, Enigma, Pi and Good Will Hunting. I offer my own analysis of the films, and make connections to the teaching and learning of mathematics. In particular, I argue that the films create gendered pictures of what being a mathematician and doing mathematics mean, and that these pictures have powerful impacts on the ways in which learners' construct their relationship with the subject.
10 Linking multiple representations in exploring iterations: does change in technology change students' conjectures?
This study investigates changes in conjectures of four typical students when they are using different kinds of technologies, particularly in relation to their preferences for representations and the way they express their conjectures in understanding the concept and properties of iteration. The first stage of the research was conducted using pen and paper (PP) with graphical calculator (GC) in a classroom while the second stage used PP with graphical software (GS) in a laboratory. The findings suggest, with important caveats, that different technologies significantly influence the students' preferences for representations. Also, this study shows that students' conjectures can be an effective unit of analysis in researching students' understanding of iteration and preferences for representations.
11 Objectives driven lessons in primary schools: cart before the horse?
Advice to share learning outcomes with pupils may be based on sound theoretical and practical principles. However, in order to turn intended outcomes into classroom experiences teachers have to draw on their interpretations of objectives and their mathematical pedagogic content knowledge. In this paper I argue that this may place non-trivial demands on primary school teachers and requires a subtle understanding of the mathematics involved.
In this paper we outline the policies that have created parallels between numeracy work in schools and with adults. A 'one size fits all' pedagogical and curriculum stance has led to an adult numeracy curriculum which is very largely based on the national numeracy strategy; we discuss potentially contradictory issues in the field of adult education.
In this article we use data from two mathematics departments within one local education authority implementing a national reform policy – the mathematics strand of the Key Stage 3 (KS3) Strategy – to explore the contrasts in the interrelationships between the views of, and goals for, mathematics teaching and learning that teachers see within the policy compared to their own views and priorities. The ways in which these contrasting interrelationships in views and goals impact upon the profile of the department in the context of policy implementation are considered.