Eisenberg, T., Engelbrecht, J. & Mamona-Downs, J. (2010). Advanced mathematical topics: transitions, evolutions and changes of foci

As Zazkis & Holton observe about recent studies of transition from school to university mathematics examine ‘issues of transition as related to both curriculum and pedagogy’ (2005, p129). With reference to papers included in the IJMEST SI on the teaching and learning of undergraduate mathematics, they stress:

‘The issue of challenges in transition to undergraduate and advanced mathematics is of significant concern to the mathematics education community. Lovric points to a trend that school graduates are less prepared in dealing with university level mathematics using the result of the recent imposed change in the province of Ontario, Canada. Luk provides a personal account of transition, describing the challenges he faced both as a student and as teacher of undergraduate mathematics in Hong Kong. Hockman presents a concern of “watering down” courses in order to comply with the need to accommodate a larger amount of students and lack of support from administration in South Africa.

These papers raise a universal concern - the concern of deterioration. This goes hand in hand with what Selden refers as “two contradictory trends”: the advocacy for school graduates who are better prepared mathematically for both university and the work place, and the seeming desire by legislatures and administrative bodies to reduce levels of certification.’ (p129)

From:

Zazkis, R. & Holton, D. (2005). Foreword. International Journal of Mathematical Education in Science and Technology, 36(2), 129-130. doi:10.1080/0020739042000196331

Similar observations offer the editors of the 2005 IJMEST SI that reported work from the DELTA conference of that year:

‘A key topic at all the Delta conferences has been the transition from secondary to tertiary mathematics education and we begin with papers on this theme from three countries: South Africa, Australia, and New Zealand. The first paper, by Engelbrecht, Harding and Potgieter, challenges the common preconception that a high school education with a focus on procedural tasks leads to undergraduate students who have trouble dealing with conceptual problems. They further explore the confidence that students have when approaching conceptual tasks. This theme is continued by Carmichael and Taylor who describe a study of students in a bridging mathematics course which indicates that student confidence contributes significantly to performance, even after accounting for prior knowledge. Barton, Chan, King, Neville-Barton and Sneddon have looked at students in Auckland for whom English is an additional language. Despite often having stronger backgrounds in mathematics, these “EAL students struggle with their learning of mathematics in English at undergraduate level much more than has been appreciated.” The paper by Oates, Paterson, Reilly and Statham looks at an effective programme of tutor training and collaborative tutorials which may help address many of these bridging issues, as well as developing a new generation of educators.’

From:

(2005). Foreword. International Journal of Mathematical Education in Science and Technology, 36(7), 699-700.doi:10.1080/00207390512331391551

Below are summaries of the selected 14 IJMEST papers.

Internationally, the consequences of the ‘Mathematics problem’ are a source of concern for the education sector and governments alike. Growing consensus exists that the inability of students to successfully make the transition to tertiary level mathematics education lies in the substantial mismatch between the nature of entrants’ pre-tertiary mathematical experiences and subsequent tertiary level mathematics-intensive courses. This paper reports on an Irish study that focuses on the pre-tertiary mathematics experience of entering students and examined its influence on students’ ability to make a successful transition to tertiary level mathematics. Brousseau's ‘didactical contract’ is used as a tool to uncover and describe the contract that exists in two case mathematics classrooms in Irish upper secondary schools (Senior Cycle). Although the authors are professional mathematics educators and well informed about classroom practice in Ireland, they were genuinely surprised by the very restrictive nature of this contract and the damaging consequences for students’ future mathematical education.

In Clark and Lovric (Suggestion for a theoretical model for secondary–tertiary transition in mathematics, Math. Educ. Res. J. 20(2) (2008), pp. 25–37) we began developing a model for the secondary–tertiary transition in mathematics, based on the anthropological notion of a rite of passage. We articulated several reasons why we believe that the educational transition from school to university mathematics should be viewed (and is) a rite of passage, and then examined certain aspects of the process of transition. Present article is a continuation of our study, resulting in an enhanced version of the model. In order to properly address all aspects of transition (such as a number of cognitive and pedagogical issues) we enrich our model with the notions of cognitive conflict (conceptual change) and culture shock (although defined and used in contexts that differ from the transition context, nevertheless, we found these notions highly relevant). After providing further justification for the application of our model to transition in mathematics, we discuss its many implications in detail. By critically examining current practices, we enhance our understanding of the many issues involved in the transition. The core section ‘Messages and implications of the model’ is divided into subsections that were determined by the model (role of community, discontinuity of the transition process, shock of the new, role of time in transition, universality of transition, expectations and responsibilities, transition as a real event). Before making final conclusions, we examine certain aspects of remedial efforts.

The transition from school to tertiary study of mathematics comes under increasing scrutiny in research. This article reports on some findings from a project analysing the transition from secondary to tertiary education in mathematics. One key variable in this transition is the teacher or lecturer. This article deals with a small part of the data from the project–analysing secondary teachers’ and lecturers’ responses to questions on the differences they perceive between school and university and the importance of calculus, a bridging content. The results provide evidence of similarities and differences in the thinking of teachers and lecturers about the transition process. They also show that each group lacks a clear understanding of the issues involved in the transition from the other's perspective, and there is a great need for improved communication between the two sectors.

Novice students at the university encounter many difficulties, linked with the secondary–tertiary transition. But what does ‘transition’ mean exactly? We consider it here from an institutional point of view, which leads us in particular to distinguish between two types of transition. We propose a specific perspective, and apply it to the case of duality in linear algebra. After describing the structure of the mathematical content concerning this theme, we discuss a survey that we have developed for the follow-up of knowledge and difficulties of students enrolled in first-year university mathematics or physics programmes, concerning duality. We present its results categorizing students’ difficulties. We explain why it is possible to interpret students’ difficulties with duality in terms of transition.

The transition process to advanced mathematical thinking is experienced as traumatic by many students. Experiences that students had of school mathematics differ greatly to what is expected from them at university. Success in school mathematics meant application of different methods to get an answer. Students are not familiar with logical deductive reasoning, required in advanced mathematics. It is necessary to assist students in this transition process, in moving from general to mathematical thinking. In this article some structure is suggested for this transition period. This essay is an argumentative exposition supported by personal experience and international literature. This makes this study theoretical rather than empirical.

Given the recent radical overhaul of secondary school qualifications in New Zealand, similar in style to those in the UK, there has been a distinct change in the tertiary entrant profile. In order to gain insight into this new situation that university institutions are faced with, we investigate some of the ways in which these recent changes have impacted upon tertiary level mathematics in New Zealand. To this end, we analyse the relationship between the final secondary school qualifications in Mathematics with calculus of incoming students and their results in the core first-year mathematics papers at Canterbury since 2005, when students entered the University of Canterbury with these new reformed school qualifications for the first time. These findings are used to investigate the suitability of this new qualification as a preparation for tertiary mathematics and to revise and update entrance recommendations for students wishing to succeed in their first-year mathematics study.

This article reports on a study carried out to measure the mathematical literacy of a selection of students entering third-level education in Ireland. The study investigates how such students performed when confronted with mathematical tasks, which, though commensurate with their level of education, may not have been familiar to them, and to identify the factors influencing their performance. Moreover, the relationship between the skills measured by the test of mathematical literacy administered and those measured by state examinations was explored, as was the question of whether or not the concept of mathematical literacy is a useful one for third-level educators.

Hunt and Lawson 1 displayed the evidence of decline in the mathematical standards of first-year students in Coventry University between 1991 and 1995. Gill sought to investigate if this was also the case in the University of Limerick (UL). The results of diagnostic tests administered to first-year undergraduates in the science and technology groups (service mathematics courses) between 1997 and 2002 displayed the evidence that the mathematical standard of students entering the university of Limerick service mathematics courses had declined over the 6 years studied. In this article, the authors revisit the university of Limerick database, which currently holds data for over 6200 students, to investigate current mathematical entry standards of students in service mathematics courses. The university of Limerick responses to the ‘Mathematics Problem’ are also described. Ireland presents a unique situation in terms of the mathematical homogeneity of its third-level students; most students enter via the same route, i.e. the Leaving Certificate, on completion of 13 years of formal mathematics education. However, while research results and coping mechanisms in terms of learning support are not generalizable, they are portable. It is hoped that mathematics educators worldwide can learn from the Irish situation.

An increasing number of Australian students elect not to undertake studies in mathematical methods in the final years of their secondary schooling. Some higher education providers now offer pathways for these students to pursue mathematics studies up to a major specialization within the bachelor of science programme. This article analyses the performance in and engagement with mathematics of the students who elect to take up this option. Findings indicate that these are not very different when compared to students who enter university with an intermediate mathematics preparation. The biggest contrast in performance and engagement is with those students who have studied mathematics in senior secondary school to an advanced level.

It seems self-evident that there is a significant gap between secondary school and university mathematics, even though the gap may take different forms, which vary with different education systems in different places and at different times. This paper attempts to capture some common forms of this gap, and in particular, it discusses some core factors that are directly related to the nature of mathematics. Further, it also attempts to find ways to bridge the gap more easily and surely for students.

During the years they spend in university, many mathematics students develop a very poor conception of mathematics and its teaching. This fact is bad in all cases, but even more in the case of those students who will be mathematics teachers in school. In this paper it is argued that the history of mathematics may be an efficient element to provide students with flexibility, open-mindedness and motivation towards mathematics. The theoretical background of this work relies both on recent research in mathematics education and on papers written by mathematicians of the past. Opinions are supported with examples. One example concerns a historical presentation of ‘definition’; it was developed with mathematics students who will become mathematics teachers. For students oriented to research or to applied mathematics, an example is presented to address the problem of the secondary-tertiary transition.

Research on transition issues and more globally on teaching and learning processes at the transition between high school and university or in the first university years has been addressed in this community since the early eighties (cf. for instance the doctoral thesis by Robert on the notion of limit in 1982, research on integral and differential processes in maths and physics education by Artigue, Legrand, Viennot and other colleagues, research on the teaching of differential equations by Artigue and Rogalski, in the eighties, research on linear algebra by Dorier, Robert, Robinet and Rogalski leading to a book published by Kluwer in 2000 in the nineties, all this research being referenced for instance in the ICMI Study devoted to the teaching and learning of mathematics at university level). Even if we focus in this survey on research published in the last ten years, there is no doubt that most approaches and theoretical constructs used today emerged more than ten years ago. For instance, the idea of FUG concept (FUG for Formalizing, Unifying, Generalizing) emerged in the research carried out in linear algebra mentioned above, the institutional approach to transition processes inspired by ATD (Anthropological Theory of Didactics due to Chevallard) emerged in Grugeon’s thesis (1995) regarding transition between general and vocational secondary education and was imported into the study of the secondary-university transition by Praslon in his doctoral thesis (2000).

An approach to this set of research work can be organized in different ways. The information that we received on the survey encourages an organization by mathematical domains. But for understanding the affordances and specificities of Francophone research about these transition issues, for instance the importance attached to epistemological and mathematical analyses, or the increasing influence of ATD, a transversal vision seems also useful. We will try to combine them. Good reference for such a transversal view is also (Gueudet, 2008), (Artigue, Batanero, Kent, 2007) even if these are not restricted to the Francophone community.

References

Artigue M., Batanero C., Kent P. (2007). Learning mathematics at post-secondary level. In F. Lester (ed.), Second Handbook of Research on Mathematics Teaching and Learning, 1011-1049. Information Age Publishing, Inc., Greenwich, Connecticut.

Gueudet G. (2008). Investigating the secondary-tertiary transition, Educational Studies in Mathematics, 67-3, 237-254.

Holton D. (Ed.) (2001). The teaching and learning of mathematics at university level. A ICMI Study, 207-220. Dordrecht : Kluwer Academic Publishers.