A report on an experience in comparing theoretical approaches with respect to the construction of research problems

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A report on an experience in comparing theoretical approaches with respect to the construction of research problems
   1 How are theoretical approaches expressed in research practices? A report on an experience in comparing theoretical approaches with respect to the construction of research problems   Susanne Prediger IEEM - Institute for Development and Research in Mathematics Education, University of Dortmund, Germany, prediger@math.tu-dortmund.de To appear in ZDM - The International Journal on Mathematics Education 40 (2008) 2. The srcinal publication is available at www.springerlink.com.   Abstract  This article explores the idea that theoretical approaches might be usefully compared in terms of the ways in which they lead researchers to construe commonsense classroom problems. It reports an ex- perience when one such problem was posed to a range of researchers with different theoretical back-grounds. They were invited to propose an answer, and to reframe the classroom problem as a research  problem. As anticipated, responses adopted particular theoretical perspectives that ‘privileged’ certain objects of study and modes of explanation. Nevertheless, where responses did appeal to a common theo-retical perspective, sometimes used in combination with others, there could be quite sharp differences in conceptualisation, proposed action, and research intention.   How should the scientific community in mathematics education deal with the diversity of theo-retical approaches within the field? Rather than the frequent demand for unifying  theories, an increasing number of researchers plead for the primacy of comparing and    understanding the differences and commonalities of different theories (e.g. Artigue et al., 2006; Bikner & Prediger, 2006; Cobb, 2007; Lerman, 2006). This process of understanding different theoretical ap- proaches has always been an important part of the disciplinary discourse, e.g. during CERME conferences. Given its complexity and the richness of different theories, it is far from being finished (see Artigue et al.; 2006, Arzarello et al.; in press). The general plead for comparisons raises the question for concrete criteria, methods and focuses for these comparisons. This article follows a suggestion made by Paul Cobb (2007) in his recent handbook article on “coping with multiple theoretical perspectives”, namely to “compare and contrast various perspectives by using as a criterion the manner in which they orient and con-strain the types of questions that are asked about the learning and teaching of mathematics, the nature of the phenomena that are investigated, and the forms of knowledge that are produced” (Cobb, 2007: 3). Also Bergsten (2007) raises the question “How does a theoretical basis chosen for a study influence the nature of the purpose, questions, methods, evidence, conclusions, and implications of the study?” The  focus on the expression of theoretical approaches in research practices  as proposed by these authors can be understood in the light of Charles Sanders Peirce’s pragmatic maxim: “In order to ascertain the meaning of an intellectual conception one should consider what  practical consequences  might conceivably result […from it]; the sum of these consequences will consti-tute the entire meaning of the conception.” (Peirce, 1905: CP 5.9). This article reports on an experience (Prediger & Ruthven, in press) we made while exploring (in line with Cobb’s suggestions) one practical way to compare different theoretical approaches   2in terms of the ways in which they construe issues of classroom practice (we called it “teaching  problems”), and propose in which way these can be researched. Other authors in this ZDM-issue compare how differences of theoretical approaches influence the ways of analysing given data (such as Gellert, Halverscheid or Maracci). They show striking effects which help to understand the field. However, all these articles start their comparison at a rather late stage of the research process, although it is already widely recognised that preference for a particular theory will already in-fluence the way in which researchers construe a problem and investigate it (Silver & Herbst, 2007; Schoenfeld, 2007; Lester, 2005; Sriraman & English, 2005/2006). Many decisions have already been taken by the time that we come to the data analysis, including: •   initial identification of a problem in classroom practice, loosely framed; •   conceptualisation of the classroom problem; •   transformation of the problem into more focused research questions; •   development of research design (incl. methodological choices, sample…). Therefore, we decided to start earlier in the research process by considering how specific theo-ries shape the (re)formulation of an initially loosely framed problem of professional practice. By doing this, we reflect the emphasis given by Schoenfeld (2007: 23ff) on the conceptualisation of a situation as a crucial step in the research process. To sum up, the following questions guided our experiment: •   How are theoretical approaches expressed in the practice of researchers, especially in the steps from a loosely framed classroom problem to the construction of a research problem and development of a research design? •   What similarities and differences can we find when comparing different theoretical ap- proaches? •   In how far are the research practices determined by theoretical choices? •   What methods can we use to treat these comparing questions? This article cannot give an extensive and systematic answer to the last question on methods for comparison. But it can report on an interesting experience we (Kenneth Ruthven and myself, see Prediger & Ruthven, in press) made with an experiment which was srcinally designed to initi-ate discussion in the ERME working group on theoretical approaches before and while CERME 5 (see also Arzarello et al., in press for the whole group’s work). Although the experiment should not be mistaken as a systematic research study, it might give some ideas about how con-solidated research methods could evolve from these first experiences.   3 1. The approach in the experiment We asked researchers with different theoretical backgrounds to briefly describe, first how they would conceptualise a given teaching problem, and then how they would design an appropriate research study. The initial reference point was a classroom problem, which we chose since we have often heard it expressed along the following lines:  How is it that some students can learn to tackle a particular type of mathematical problem successfully (as shown by their performance in the class), but be unable to do so two weeks or months later? What strategies can the teacher use to reduce the likelihood of this occurring? The questions themselves presuppose certain assumptions which are not necessarily shared by all researchers, namely the idea that a teacher might be able to reduce the likelihood of the prob-lem occurring, and that research should transcend purely descriptive analytical purposes and include reflections on strategies for changing teaching practices. In order to see how different approaches frame this classroom problem as a research problem and devise a research design, we asked the following questions: a) How do you –a priori– answer this question and what are your basic assumptions? b) How do you transform the raised problem into a research question starting from the ques-tion above? c) What is your research design? d) What type of results would you expect? The questions were sent to 34 European researchers who demonstrated an interest in our re-search questions as they had registered for the working group on theories on CERME 5 (see Arzarello et al.). As a whole, we got eight answers from 14 researchers, which are completely  printed in the appendix to Prediger and Ruthven (in press). As some of the responses were too vague for a deeper analysis and for reasons of pragmatic restriction, we chose only 5 of the re-sponses for this article, namely those given by Michèle Artigue and Agnès Lenfant, Ferdinando Arzarello and Ornella Robutti, Marianna Bosch and Josep Gascòn, Tommy Dreyfus and Ivy Kidron, and Helga Jungwirth. The researchers had a time restriction of 15 days to answer, and they were asked to restrict to 2-3 pages. Hence, the given responses were not completely spontaneous, but should of course not  be mistaken as mature research papers. We consider them as self-presentations of snapshots from research practice, in all its tentative form. As their responses do not strictly follow the questions and have different grades of explicitness in different aspects, the tentative analysis presented in the following three sections is not in-tended to completely meet quality standards for empirical studies like interrater reliability. Nev-ertheless, it gives an impression on interesting differences and commonalities.   4 2. An initial survey of the responses How do the different authors conceptualise the given classroom problem and reframe it as a research problem? Most of the responses accept to a degree the srcinal terms in which the  problem was posed, but suggest that these alone are inadequate to frame it. Equally, while most of the responses recognise that a range of factors may play a part, and that different lines of explanation can be developed, each adopts a particular theoretical perspective which privileges certain objects of study and modes of explanation. This section gives an initial survey of the responses with longer quotations and orientational comments. By this, we hope to make the researchers speak themselves, although risking that not all arts are easy to understand.  Artigue and Lenfant Artigue and Lenfant suggest that the srcinal teaching problem is “a rather banal phenomenon”: “[W]hat we learn is most often not definitively learnt, and if we do not use what we have learnt, generally, more or less quickly we forget it.” However, Artigue and Lenfant argue that this way of thinking about the problem “does not have a specific didactic flavour and could lead to look for explanations only at the level of the brain functioning or at the level of personal motivation for studying such or such topic, for learning to solve such or such type of task… [whereas a] didactic approach offers alternative or complementary perspectives, and will not necessarily lead to the same suggestions for improving the situation.” Accordingly, Artigue and Lenfant articulate a preference for explanations (and interventions) which frame the teaching problem in distinctively didactical terms. Due to their theoretical framework, these didactical terms should be specifically mathematical: “There is certainly a lot of literature about such issues in cognitive research. From a didactic  perspective, what seems more interesting to us is to transform the raised problem into a research question in such a way that the specificity of mathematics knowledge, of mathematical and di-dactical organizations could be taken into account, and that a systemic view could be developed, the ‘forgetting student’ being no longer the exclusive or central object of our attention.” Artigue and Lenfant, proceed, then, within a theoretical framework drawing on the Theory of Didactic Situations [TDS] (see Brousseau, 1997) and the Anthropological Theory of Didactics [ATD] (see Chevallard, 1992), on the basis that “the observed phenomenon [of forgetting], if not created, is highly reinforced by [a wide range of] didactical choices” concerning treatment of the task and organisation of the task environment: “how this type of task was introduced to the students with what mathematical motivations, how techniques for solving it were developed, how did the respective responsibilities given to the students and the teacher in the solving of this type of task progressively evolved, up to what  point some particular techniques were trained and routinized, how the variation around this type of tasks was organized taking into account its didactic variables, up to what point the mathe-matical knowledge at stake was explicitly pointed out, justified, institutionalized and how the necessary decontextualization of knowledge was worked out, how this type of task was related with other ones in wider mathematical organizations, what opportunities were given to make the   5students’ relationship with this task evolve beyond the necessarily short period of its official teaching.” Their possible research questions refer to the wide range of questions “orientated towards … understanding … [and] didactical engineering trying to improve the current situation”; and they give examples for general question which would be concretized for specific research projects: “Q1: Are different types of mathematical tasks equally sensitive to the ‘forgetting phenome-non’ and what can explain observed differences if any? Q2: What are the strategies that mathematics teachers tend to use for limiting or controlling the ‘forgetting phenomenon’? What is the rationale underlying these and what are their effects? Q3: Are there characteristics of the usual mathematical organizations which tend to reinforce the ‘forgetting phenomenon’ and, if so, what are the mechanisms underlying this rein-forcement? Q4: Does an engineering design where specific attention is paid to the balance between the different moments of the study (according to the TAD) and to the completeness of mathematical praxeologies can make a difference?” To sum up, Artigue and Lenfant present a clear reframing of the classroom problem in didacti-cal terms that refer to the specificity of mathematics.  Arzarello and Robutti Arzarello and Robutti characterise the srcinal teaching problem as “a natural fact”, framing their discussion in terms of the distinction that the Anthropological Theory of Didactics [ATD] (see Chevallard, 1992) makes between technique and theory: “Maybe that a person many years after she ended the school remembers something about the theories but has forgotten everything concerning the techniques… and so is not able to solve the  problem… Maybe a ‘feeble’ student remembers the technique but not the technology and the theory: so she is not able to solve the problem for different and opposite reasons. It is a question of level at which the knowledge related to the problem must be known to solve it. It is clear that without a continuous training many abilities linked with techniques and technologies become lower. This may cause lower performances and is a natural fact. Of course this depends on the type of performances asked and on the level of assimilation of the techniques, technologies and theories required by the performance itself. Hence to tackle the question the teacher must distin-guish carefully at which level the performances of a task are situated.” In this light, Arzarello and Robutti identify three “key variables”, namely “different specific mathematical contents”, “the level …at which the performances for a specific knowledge …in the task are required, [and] the methodology of teaching”. Their research interest is focused on different methods of teaching which are related to some underlying theorisation: “A) a traditional approach, based on the sequence: explanation-exercise-repetition-assessment; B) a more innovative approach, where the knowledge is constructed by students in suitable learning situations, based on the use of laboratory and ICT. These two approaches can be analysed according to the different ways of teaching-learning they  produce from a cognitive point of view. For this, two related types of analysis can be developed,
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