Icme 12 Survey Report Key Mathematical Concepts in the Transition from Secondary to University Background
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- Transition papers published in IJMEST
AbstractThis article has as a purpose to introduce, briefly, a Phenomenological investigation about the construction knowledge of the Algebra structures, which moves around the Mathematics Philosophical Education focused on the question: how is the thinking shown in the movement of the knowledge construction of the Algebra structures? There are in this article: the research mainly question, the investigative movement and the analyses development; focused on the way the research ideas were built, because they build the research and its implication on the Mathematics Education about the way the Abstract Algebra is taught. Key-words: Algebra. Hermeneutic. Phenomenological Cogitation. Mathematics Philosophical Education.
Publicação Semestral da Faculdade de Educação da Universidade Estadual de Campinas (Unicamp) - Campinas (SP) Brasil em Educação Matemática ZETETIKÉ volume 17, no. 31 (2009) 6 - Praxeologías Didactic in the University: A study of case about the notions of Limit and Continuity of functions p.151-190 Parra Verónica, Núcleo de Investigación en Educación en Ciencia y Tecnología Maria Rita Otero, Facultad de Ciencias Exactas Abstract This work is a case study based on the Mathematical Analysis applied to Economy and Administration in the first university level in this area. The didactic praxeology of a university teacher is described using the Anthropological Theory of Didactics (ATD) (Chevallard; 1992, 1997, 1999, 2000). The restrictions and exigencies that limit the educational practice in the University are analyzed. The didactic phenomenon of autism defined by Chevallard (1999) is described and an autism related to the exams is identified. The university teacher’s difficulties to resolve the didactic problems around the concepts of limit and continuity of functions are analyzed. Keywords
Anthropological Theory of Didactics. Didactic Praxeology. University level. Limit and Continuity of functions. Autism. Corriveau, C. (2009). Formalisme et démonstration en algèbre linéaire. EMF 2009. Groupe de travail 7. Dakar INTRODUCTION In Quebec, the transition from secondary level to university is characterised by an increased responsibility of students concerning organisation, more complex notions to be learnt but also more demand in mathematical rigor. The questions asked are:
1.1. New practices expected from students
1.2. The formalism obstacle
The analysis has been done in two phases: the a priori analysis of the tasks’ complexity according to Robert’s framework; the analysis of students´ answers to these tasks. This analysis allows looking more precisely at students’ difficulties.
Task 1 – a proof using a definition (see p. 4) – was classified as simple. Task 2 – an unusual proof (see p. 5) – was classified as complex.
11 to 12 works from a group of students. Analysis of 2 of them for the tasks 1a and 2 (see p. 6-8)
Several students’ difficulties and errors which come at the same time from the manipulation of objects in linear algebra and more generally from proof.
The 2 tasks had been a priori evaluated, respectively as simple and complex. The productions’ analyses show that for students they were respectively complex and very complex. The a priori analysis of tasks is necessary to measure the task’s complexity. The study of students’ productions complements this analysis and gives more precisely the level of complexity. The author suggests a new formulation of the 2 tasks, taking into account the difficulties that have been identified in order to choose intermediate questions and the values of the didactic variables (see p. 11-12). CONCLUSION The formalism obstacle appears when students work with expression loosing sight of the mathematical objects that the symbols represent. One of the challenges of the transition secondary level – postsecondary level is the « learning a new algebra paradox ». A new algebra (elementary algebra, vectorial algebra, matrix algebra, etc.) is introduced as a tool for calculation, for « automatisation », for « algorithmisation » of procedures, of reasoning through calculations and their rules. This means that we accept to delegate parts of the control of validity and meaning to this algebra. But this also leads to loosing control and meaning. Drouhard’s hypothesis (2006): concerning transitions, the changes of the mathematical game’s rules are obstacles much more important that the simple extension and deepening of the mathematical objects’ domain of study.
Several papers presented at CERME7 WG14 (which included issues of transition in its Call for Papers) focused on the transition from school to university mathematics. Below I summarise these papers. I then list the full reference for each at the end of this section. Gyöngyösi, Solovej & Winslow employs Chevallard’s Anthropological Theory of Didactics to describe a transitional course in Analysis that was taught with a combination of Maple and paper-based techniques and resulted in mixed reception and performance by students. Of particular importance is the way in which this paper employs a combination of theoretical frameworks to study transition: an adaptation of Chevallard's ATD by Winslow (the study of mathematical praxeology in terms of the 4T: tasks, techniques, technologies and theories), Artigue's notions of epistemic and pragmatic value, Winslow's semiotic representation, the construct of instrumental genesis and Trouche's instrumental orchestration. The collected data include sets of tasks, student coursework and student evaluation forms The paper is a solid example of the use of ATD theory in university mathematics education (here a study focusing on new students' transition to more formal aspects of Analysis - series - through the use of CAS). It provides clear definitions of the constructs employed in the research. Even though written in an extensively symbolic language in its use of these constructs, the points about how the use of instruments changes the kinds of mathematics students do come across clearly. The paper is less confident in the way it treats the data from student evaluations and overall the recording of the student experience. It leaves the impression that ATD offers a comprehensive theoretical framework for exploring certain aspects of the student experience (epistemological, cognitive and institutional). Stadler describes students’ experience of the transition from school to university mathematics as an often perplexing re-visiting of content and ways of working that seems simultaneously familiar and novel. The mathematical focus here is on students’ work on solving a parametric system of simultaneous equations and the difficulties they experience with working with variables, parameters and unknowns. Data include student observations and interviews and the perspective is discursive and enculturative, largely Sfard’s commognition. The paper illustrates the multi-faceted nature of transition from school to university mathematical discourse through the extensive examination of a selected episode. While the analysis of the episode as a case that illustrates several facets of the transition (individual, institutional, social) is not totally convincing (the students' difficulty with variables, parameters and unknowns is palpable and slightly overshadows the other aspects), this is overall a neat and apt application of Sfard's perspective. Throughout the impression is that Sfard's perspective is a good match for studies of transition. Nardi makes similar use of Sfard’s perspective. In interviews, university mathematicians comment on newly arriving Year 1 students’ verbalisation skills and note: the role of verbal expression to drive noticing; the importance of good command of ordinary language; the role of verbalisation as a semantic mediator between symbolic and visual mathematical expression; and, the precision proviso for the use of ordinary language in mathematics. One observation that emerges from the analysis is that discourse on verbalisation in mathematics tends to be risk-averse. Linked to this is the observation that more explicit, and less potentially contradicting, pedagogical action is necessary in order to facilitate students’ move away from often wordless mathematical expression in school and appreciation of mathematical eloquence. The examples that the interviewees touch upon span across mathematical topics but the paper focuses on examples from Group Theory and Linear Algebra. The paper draws on a larger pool of data presented in (Nardi, 2008). Biehler, Fischer, Hochmuth & Wassong proposes that blending traditional course attendance with systematic e-learning study can facilitate the bridging of school and university mathematics. Their data include student evaluation data (questionnaires) and explore a range of personal and institutional variables. Early indications of findings point favourably towards the e-learning part of the course. Faulkner, Hannigan & Gill note the intensely shifting profile of students who take service mathematics courses. Specifically they report that between 1998 and 2010 the profile of students who take service mathematics courses in the University of Limerick (Ireland) has changed dramatically: many more are diagnosed as at risk, fewer have an advanced mathematics secondary qualification and the percentage of non-standard (e.g. mature) students has grown (but is also the type of student better improving performance due largely to systematic use of newly offered types of learning support). Their data originate in student profiles and a diagnostic test database This paper adds to the intrernational body of evidence on changing student profiles in the transition from secondary to tertiary education. It also records a set of systematic responses to this changing profile in the particular institution. It is not a research paper as such but its longitudinal database has the potential to lend itself to substantial analysis and reflection. See also related IJMEST paper. Vandebrouck noted that the transition from school to university mathematics students need to reconceptualise the concept of function in terms of its multiple registers and its process-object duality. The theoretical approach taken includes a collection of concept-image and process-object related theoretical constructs with some reference to Tall's three worlds. Data are mostly student scripts. While the idea of approaching issues of transition from school to university mathematics from the perspective of the transition from the embodied to the symbolic and formal world is a well-trodden, almost classic idea, it is these days a little schematic. Other more institutionally etc. sensitive perspectives are rather richer ways to examine transition. De Vleeschouwer & Gueudet observe that students can learn to appreciate the duality in linear forms (process-object or, to these authors, micro-macro) if given an appropriate set of tasks that require them to engage with these concepts at both levels. This paper also draws on Brousseau's didactical contract and Chevallard's 4T framework and its data are largely student scripts. The paper revisits classical ideas, such as the process-object duality of the linear forms in Linear Algebra, in order to put forward the point that some of the difficulties that the students experience may originate in the institutional experiences they have been offered (e.g. tasks). The perspective here is that of the changing didactical contract between school and university mathematics, particularly with regard to ways of approaching mathematical content (and less of the more common in research foci on more general aspects of the students' mathematical learning experiences such as teacher expectations, attitudes to proof etc.) Iannone & Inglis discuss a range of weaknesses in newly arriving Year 1 mathematics students’ production of deductive arguments (rather than in the oft-reported perception that a deductive argument was expected of them). Specifically, Year 1 mathematics students responded to four proof tasks and demonstrated a range of weaknesses in their production of deductive arguments. The data comes from a larger pool of data (student responses to self-efficacy (not used here) and proof survey (4 proof tasks). The paper offers evidence that newly arriving Year 1 mathematics students are aware that when asked to generate a proof, they are asked for a deductive argument. This is in some contrast to previous work in the field but often this contrast may be accounted for by different student background and specialisms in the student sample. Above papers will be published soon in the CERME7 Proceedings as follows: Biehler, Fischer, Hochmuth & Wassong (2011, in press). Designing and evaluating blended learning bridging courses in mathematics. In Swoboda, E. (ed) Proceedings of the 7th Conference of European Researchers in Mathematics Education (pp. tbc-tbc).Rzeszow, Poland. Faulkner, F., Hannigan, A. & Gill, O. (2011, in press). The changing profile of third level service mathematics in ireland and its implications for the provision of mathematics education (1998-2010). In Swoboda, E. (ed) Proceedings of the 7th Conference of European Researchers in Mathematics Education (pp. tbc-tbc).Rzeszow, Poland. Gyöngyösi, E., Solovej, J.P. & Winslow, C.(2011, in press). Using CAS based work to ease the transition from calculus to real analysis. In Swoboda, E. (ed) Proceedings of the 7th Conference of European Researchers in Mathematics Education (pp. tbc-tbc).Rzeszow, Poland. Iannone, P. & Inglis, M. (2011, in press). Undergraduate students’ use of deductive arguments to solve “prove that…” tasks. In Swoboda, E. (ed) Proceedings of the 7th Conference of European Researchers in Mathematics Education (pp. tbc-tbc).Rzeszow, Poland. Nardi, E. (2011, in press). ‘Driving noticing’ yet ‘risking precision’: University mathematicians’ pedagogical perspectives on verbalisation in mathematics. In Swoboda, E. (ed) Proceedings of the 7th Conference of European Researchers in Mathematics Education (pp. tbc-tbc).Rzeszow, Poland. Souto-Rubio, B. & Gómez- Chacón, I. (2011, in press). Challenges with visualization at university level: The concept of integral. In Swoboda, E. (ed) Proceedings of the 7th Conference of European Researchers in Mathematics Education (pp. tbc-tbc).Rzeszow, Poland. Stadler, E. (2011, in press). The same but different - novice university students solve a textbook exercise. In Swoboda, E. (ed) Proceedings of the 7th Conference of European Researchers in Mathematics Education (pp. tbc-tbc).Rzeszow, Poland. Vandebrouck, F. (2011, in press). Students’ conceptions of functions at the transition between secondary school and university. In Swoboda, E. (ed) Proceedings of the 7th Conference of European Researchers in Mathematics Education (pp. tbc-tbc).Rzeszow, Poland. De Vleeschouwer, M. & Gueudet, G. (2011, in press). Secondary-tertiary transition and evolutions of didactic contract: the example of duality in Linear Algebra. In Swoboda, E. (ed) Proceedings of the 7th Conference of European Researchers in Mathematics Education (pp. tbc-tbc).Rzeszow, Poland. In this section I also referenced: Nardi, E. (2008). Amongst mathematicians: Teaching and learning mathematics at university level. New York: Springer.
Note 3: The work of this team can be massively assisted by the 2010 IJMEST SI edited by Eisenberg, Engelbrecht & Mamona-Downs based on papers presented at ICME11 in Mexico. I cite below the Editors’ Foreword to this and include summaries of the relevant papers in Section 3: ‘The 10 papers published in this special issue share a common root in that they all originate from talks addressed at the same group, Topic Study Group 17, of the eleventh International Congress on Mathematical Education held at Monterrey, Mexico, 2008. The papers that appear here are substantial extensions, refinements or re-workings of the presentations that were originally published on the conference proceedings website. The original title of the theme of the group was: 'Research and development in the teaching and learning of advanced mathematical topics'. The 'advanced' was taken to refer to mathematical material from the final two years of secondary schooling and beyond, but was mainly levelled at undergraduate study. Papers were welcomed whether they dealt with specific 'mathematical topics' or addressed issues running through different topics. The educational literature on this level of mathematics teaching and learning is now quite extensive. Here we organize our summaries of the papers in such a way to give the collection some distinct identity. Several of the papers stressed facets of transition, but in rather different contexts. A transition usually indicates a state in learning for which there are inherent difficulties to overcome from a previous one. On the other hand, without employing the term explicitly, several papers point to aspects of 'evolution', i.e. the assimilation of mathematical practice or conceptualization over a considerable time. Finally, much of the doing of mathematics concerns different sources of thinking that are available at a certain time and have to be co-ordinated; this largely concerns a willingness to affect 'changes of focus'. On these three 'axes', we group the papers as below. The kind of transition most written about is the perceived difference between the styles of mathematics taught at school against that at university. In his essay, Engelbrecht gives a wide vista on what working in advanced mathematics involves; the need for logical structure is stressed, indeed sometimes it constitutes the only way to proceed, but mental processing, conceptualization and intuition also have a crucial, complementary role. He considers formal presentation and verbal communication vis- Other aspects of mathematical thinking of an individual seem to develop over time without the individual being fully conscious of how they evolved. Selden, McKee and Selden examine non-emotional cognitive feelings and behavioural schemas as enduring mental structures that link situations to actions. These can be beneficial in some cases, detrimental in others. Examples are given concerning students' acceptance (or not) of the notion of a fixed but arbitrary element, and the behaviour that some students show in focusing too soon on the hypotheses rather than the result. Teaching intervention in this context is also discussed and illustrated. Berg Various types of effecting changes in focus are featured in the remaining four papers. Two papers explicitly deal with representations, i.e. the use of alternative contexts that reflect the same mathematical structure at hand but allow a relaxed environment for which mental or informal argumentation to act. Arnoux and Finkel consider the responsibility of the teacher to design representations such that they become an integrated aspect of teaching. Faithful representations are rarely encountered in standard expositions of advanced mathematics topics, so it is important to build up a catalogue for instructional purposes. The paper gives examples concerning models of graphs and automata, and a mechanical model in probability. Lagrange describes a project aimed at improving secondary school students' understandings of functions and their handling. Mainly the project concerns the use of computer representations, designed on two principles, to insist on an environment allowing the linking of a dynamic geometry component with an algebraic unit, and to permit students to experiment freely. He also discusses the need for class discussion moderated by the teacher to collate the theoretical significance of the students' work. Koichu confronts the issue that in (relatively) advanced mathematics, there are cases where problem solving behaviours, such as invoking heuristics and executive control, seem to be more decisive than employing associated bodies of mathematical knowledge. Even capable students can invoke knowledge that simply impedes a more basic approach. One explanation made of this phenomenon is that students wish to take the path that, to them, constitutes the least intellectual effort. Mamona-Downs illustrates how a mathematical theory can be recast simply by taking a slightly different perspective; the intention is to broaden students' appreciation beyond what is offered by the standard way it is taught. In particular, the 'underlying' set of real sequences is considered, yielding a way to gauge the sequence limiting behaviour in terms of accumulation points. In this respect the 'order' of the terms of the sequence is immaterial, a fact that can surprise students.’ (p139-141) From:
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traces the varying cognitive weight that students place on the concept of completeness as it occurs at different places in their undergraduate career. The initial need to address an issue concerning the 'continuous' image of the number line formally leads to completeness to be taken as an axiom for which there are many equivalent formulations. The axiom enables proof of some fundamental theorems in Analysis that are simply assumed in Calculus. The paper discusses the situations where students regard