Collaboratve Conceptual Change : The Case of Recursion


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The paper tries to shed some light on the conceptual development of computer-science concepts by discussing recursion as an interdisciplinary idea and by analyzing the class discourse during an introductive learning activity. Throughout the inductive analysis of the discourse, the students’ expressions were interpreted, refined, and formulated as ‘preconceptions’. After a short theoretical and methodological background (in Secs. 2 and 3), we present these pre-conceptions, suggest an organizing model of conceptual change, and discuss certain implications concerning the issue of understanding recursion by high school students and of the collaborative development of such understanding.

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Collaboratve Conceptual Change : The Case of Recursion

  1. 1. Collaborative Conceptual Change: The Case of Recursion Dalit Levy Department of Education in Science and Technology, Technion–Israel Institute of Technology, Haifa 32000 IsraelABSTRACT The growing body of research within computer-science education hasnot yet focused on studying conceptual change, in comparison with theintensive account of conceptual change in other science domains. Forexample, although the difficulties in learning and teaching the significantconcept of recursion are often referred to, the research literature barelyaddresses the unique ways in which students relate to this interdisciplinaryconcept, the particular learners’ language concerning recursive phenomena,and the processes of conceptual change. In order to fill the gap, the studyhere presented describes a naturalistic study in six Israeli computer-scienceclasses and deals with a variety of conceptions emerged from analyzing thestudents’ discourse of recursive phenomena. The paper also suggests a modelfor organizing the conceptions in a way that might represent a collaborativeprocess of conceptual change in regard to recursion in particular, and mightalso hint at the communal nature of conceptual change in general. In terms ofthis paper, that nature is referred to as ‘collaborative conceptual change’.KEYWORDSclass discourse, computer-science education, functional programming,constructivism, qualitative research, recursion _____________________________________________________________ Reprint requests to: Dalit Levy, Department of Education in Science andTechnology, Technion–Israel Institute of Technology, Haifa 32000 Israel;e-mail: 113
  2. 2. Vol. 12, No. 2, 2002 Collaborative Conceptual Change: The Case of Recursion1. INTRODUCTION ‘Look, this one is like the other but smaller’ said Pavel while analyzingthe tree in Fig. 1. When saying ‘this one’, he was pointing at the right-mosttree and was drawing an imaginative little circle around it. When saying ‘likethe other’, he drew a much larger circle around the whole tree figure. Pavel, asixteen years old Israeli student, was trying to draw the attention of hisclassmates to a certain characteristic of a recursive phenomenon that he hadjust recognized. At that time, Pavel’s class was at the beginning of the majorpart of a functional programming (FP) course, the part that dealt withrecursion. Recursion is an interdisciplinary concept with many implicationsin programming (Hofstadter, 1979), or as Harvey and Wright (1993, p. 168)describe it: “recursion is the idea of self-reference applied to computerprograms”. The FP course taken by Pavel was the third of five curricular units, whichaltogether constructed the computer-science enhanced curriculum in the Carmelhigh school, a large school in a northern Israeli town (all names arepseudonyms). Pavel’s class—eleventh grade—was known at the Carmel school Fig. 1: One example of a recursive phenomenon.114
  3. 3. Dalit Levy Journal of Intelligent Systemsas ‘the scientific class’, a prestigious group of fifteen girls and boys. Theselearners were first exposed to recursion while participating in a constructivist(Ben-Ari, 2001) and interdisciplinary learning activity. The activity tookplace in the spacious computer-lab of the Carmel school, but all along its twosessions of two hours each, the students did not use computers but rather satin five groups around the tables in the middle of the room. At the beginningof the first session, each learner received a large sheet of paper with fourteenphotocopied examples of recursive phenomena taken from various sources(Fig. 1 was one of them). The learners were then asked to classify theexamples according to certain criteria of their group’s choice, to offer a titleof their own for each class of examples, and to find additional examples ofrecursive phenomena. The class discourse of the first session evolved assimultaneous independent group discussions, whereas in the second session aweek later, each group presented its classification and titles to the wholeclass. The discourse then took the form of a whole class discussion, in whichstudents of different groups were negotiating, arguing, and expressingdifferent kinds of understandings. Note that by the time of that collaborative learning activity (see Levy,2001 for more details), the students could indeed program simple functionsusing the functional language DrScheme (Felleisen et al., 1998), but they hadnever before been exposed to the formalism of recursive functions or to theconceptual framework of recursion. Accordingly, the students’ first experienceswith recursion were taking place throughout a learning activity in whichrecursion was widely viewed as an interdisciplinary concept, rooted ineveryday life and experience, not merely as a programming tool or acomputer-science exclusive idea. Such a collaborative and interdisciplinarylearning activity could stimulate a very rich class discourse concerning boththe specific examples of recursive phenomena and the general idea ofrecursion, when the learners express their unique way of conceiving, thinkingand understanding. During one such rich class discourse, Pavel wasdocumented expressing his own understanding. That idiosyncratic utterance,‘this one is like the other but smaller’, has later been categorized with othersimilar students’ expressions, and has been interpreted as related to theconceptions of self-similarity and gradualism. These conceptions are entitled 115
  4. 4. Vol. 12, No. 2, 2002 Collaborative Conceptual Change: The Case of Recursionin the study ‘preconceptions’, which is due to the initiative phase of thelearning process in which the students were involved. But as will bediscussed later, the complex network of preconceptions that emergedthroughout the study reflects an advanced framework of students’ thinkingabout recursion even at that initiative phase. Pavel’s utterance was chosen from an abundance of utterances, statements,and non-verbal expressions that were documented during the course of afield study focusing on high school students’ discourse of recursivephenomena, while trying to trace conceptual change with regard to recursion.Note that much of the research into the learning of recursion has taken placewithin the framework of learning to program (for example, George, 2000;Wu et al., 1998; Segal, 1995), whereas less emphasis has been placed onunderstanding how learners construct the concept of recursion as a broad,abstract, and interdisciplinary concept. In addition, all concerned seem toagree on the difficult nature of recursion for novices studying computer-science, both for college and university students and for younger high schoolstudents, but the literature hardly refers the latter. Moreover, the educationalresearch has not emphasized the learners’ voice or the learning processes, asthey show up in a natural setting, in a real class dealing with computer-science concepts (Booth, 1993 is one exception). And finally, in deepcontrast to the variety of publications in the field of conceptual change thatone can find regarding each of the scientific disciplines being learned at highschool (Duit, 2002; Soto & Sanjose, 2002), the focus on conceptual changewith regard to computer-science ideas and concepts has been left aside. The present paper tries to shed some light on these neglected aspects ofthe conceptual development of computer-science concepts by discussingrecursion as an interdisciplinary idea and by analyzing the class discourseduring an introductive learning activity. Throughout the inductive analysis ofthe discourse, the students’ expressions were interpreted, refined, andformulated as ‘preconceptions’. After a short theoretical and methodologicalbackground (in Secs. 2 and 3), we present these pre-conceptions, suggest anorganizing model of conceptual change, and discuss certain implicationsconcerning the issue of understanding recursion by high school students andof the collaborative development of such understanding.116
  5. 5. Dalit Levy Journal of Intelligent Systems2. THEORETICAL BACKGROUND When studying conceptual change in regard to recursion, one shouldtake into account both the framework within which most conceptual changestudies have been conducted and the literature on learning and teachingrecursion. The first kind of background can be thought of as a generalbackground in the established discipline of science education, whereas thesecond is more specific and is located within the relatively new field ofcomputer science education.2.1 The Focus on Conceptual Change in Science Education For almost three decades, conceptual change has been regarded as amost powerful frame for research on teaching and learning science (Duit,2002). Throughout these years, the notion that students often enter thescience classes with prior knowledge, ideas, and beliefs about the phenomenaand concepts to be taught became clear. Since this prior knowledge oftenstands in contrast with what is regarded as ‘scientific knowledge’, the generalterm ‘misconceptions’ was introduced, and the learners’ misconceptionswere thoroughly investigated (Wandersee et al., 1993), only to be left asidein favor of the less judgmental term ‘alternative conceptions’. In a papercriticizing the dominancy of the focus on misconceptions in the field ofscience education, the writers recommend to move toward tracing conceptualchange instead of locating more and more ‘wrong’ conceptions (Smith et al.,1993). Tracing conceptual change implies focusing at the cognitive and com-municative processes within which conceptual frameworks are constructed,organized, and reorganized. Conceptual change in science education has become a term denoting“learning science from constructivist perspectives” (Duit, 2002, p. 7). Aconstructivist belief is that knowledge is necessarily a product of our owncognitive acts and that we construct our understandings through ourexperiences (Confrey, 1995). In such a view, learning science means thatstudents themselves are constructing and reorganizing their own conceptualframeworks, in a “gradual process during which initial conceptual structures 117
  6. 6. Vol. 12, No. 2, 2002 Collaborative Conceptual Change: The Case of Recursionbased on … interpretations of everyday experience are continuously enrichedand restructured” (Vosniadou & Ioannides, 1998, p. 1213). The constructivistscience teacher should encourage a reflective discussion to expose thelearners to new conceptions and to different ideas offered by others, inaddition to her or his responsibility for offering generalizations and formalterminology. In other words, the teacher’s role is to navigate the discussiontoward creating a ‘taken-as-shared’ meaning for the scientific concepts beinglearned in the class (Cobb et al., 1992) and thus allowing some conceptualchange to occur. In this paper, conceptual change denotes pathways from students’existing conceptual frameworks to the computer science concept to belearned, which in our case is the concept of recursion. To trace conceptualchange, the researcher observed and documented students participating in aconstructivist and interdisciplinary learning activity. As the next sectionsshow, the protocols of that activity shed light both on the private conceptualframeworks students carried with them when entering the class and on aspecific kind of conceptual change which will be later described as a‘collaborative conceptual change’. A main claim of this paper is thatconstructivist learning activities like the one investigated (named CGA, seeLevy, 2001) can be characterized by a collaborative conceptual change. Inother words, although the changes occur in one’s mind and upon one’sconceptual framework, and although each student individually constructs heror his framework, conceptual change is often (always?) motivated by takingpart in more societal and communicational learning activities. In moregeneral terms, this paper (like Duit, 2002 and others) calls for mergingsociocultural views of learning within the framework of conceptual change.2.2 Learning and Teaching Recursion The literature on recursion in computer science education is wide-ranging, but page limitations allow me to mention only a few of the mostinteresting here. Hofstadter’s (1979) book Godel, Escher, Bach: An EternalGolden Braid gives a comprehensive account. Other books mainly deal withthe programming aspects of recursion (for example, Roberts, 1986; Abelson118
  7. 7. Dalit Levy Journal of Intelligent Systems& Sussman, 1996), as does most of the educational literature on recursion.Issues of learning recursion sometimes appear in textbooks together with awarning that it is not going to be easy, and teachers are often informed that teaching students to use recursion has always been a difficult task. When it is first presented, students often react with a certain suspicion to the entire idea, as if they had just been exposed to some conjurer’s trick… (Roberts, 1986, p. vii.; Wu et al., 1998; Troy & Early, 1992).These and other educational references offer several reasons for thedifficulty. Among them are the following:1. The abstract nature of the concept: “The abstraction inherent in recursion can make the process difficult for both students and teachers” (Troy & Early, 1992, p. 25).2. The different possible aspects of referring to recursion: For example, the procedure, process, and product aspects of recursion and the unequal educational emphasis put on these aspects (“The three P’s”, Leron, 1988).3. The lack of everyday analogies for recursion (Pirolli & Anderson, 1985).4. The learners’ inability to express a solution recursively and to understand the suspended computation (Bhuiyan et al., 1994; McCalla & Greer, 1993).5. The introduction of recursion after learning loop structures (Kurland & Pea, 1983), and the introduction of recursion with functions (Troy & Early, 1992). The literature suggests methods for overcoming the difficulties. Manywriters tend to agree upon one recommendation: “In order to develop a morecomplete understanding of the topic, it is important for the student toexamine recursion from several different perspectives” (Roberts, 1986, p. vii). In this spirit, Ben-Ari (1997) describes a teaching approach that stronglycouples dramatizations of simple, real-world problems with analogousrecursive programs; Harvey offers several explanations of recursiveprograms using different models (Harvey, 1985; Harvey & Wright, 1993);Astrachan (1994) declares that students should be shown as many examplesas possible for them to come to “believe” in recursion; and George (2000), aswell as Bhuiyan et al. (1994), develop a computerized environment designed 119
  8. 8. Vol. 12, No. 2, 2002 Collaborative Conceptual Change: The Case of Recursionto help students create different methods of generating recursive programs. All concerned seem to agree on the difficulty of learning and teachingrecursion. Some recommendations for teaching have been made. But whentrying to find “how it is” in a real class, in a natural educational setting, theresearch literature in computer science education is less helpful. Among thefew naturalistic accounts, one can find Booth’s phenomenographic studiesfocusing on students’ conceptions of programming, including theirconceptions of recursion (Booth, 1993), and some interview-based studiesexamining beliefs, misconceptions, and programming errors of studentslearning recursion (Segal, 1995; Lee & Lehrer, 1987). Still, the question ofwhat a computer science class looks like when it deals with recursionremains unanswered. As has been mentioned earlier, the gap is most apparent when looking atcomputer-science learning through a broader lens. When observing an inter-disciplinary and constructivist learning environment, one could also ask whatlanguage do the teacher and the students use? How can one characterize theclass discourse? How does this discourse reveal the formal aspects ofrecursion and in what ways does it expose the difficulties? What change doesthe class discourse reflects, and is it a conceptual change? And what are theaffective, social and communicational aspects involved in the conceptualchange regarding recursion? These questions have guided a naturalisticresearch on learning recursion in Israeli high schools, conducted as part ofthe author’s graduate studies during the years 1998 to 2001. In this paper, only one part of the overall research is discussed. In thispart, the focus was on the first phase of the learning process and thequestions that directed the study were the following:• What preconceptions of recursion are expressed by the learners throughout the learning activity?• What is the nature of the conceptual change in the case of that learning process? In the rest of the paper, after presenting some methodological issues, theanswers to theses question are widely discussed.120
  9. 9. Dalit Levy Journal of Intelligent Systems3. METHOD3.1 The Research Goal As briefly stated in the introduction, the research goal was to documentand analyze learners’ discourse of recursive phenomena, as a way to look atrecursion through the eyes of the learners and to help in understanding thelearners’ unique ways of speaking and thinking about the general idea ofrecursion. The formulation of such a research goal reflects twomethodological assumptions.• The first emerges from an ethnographic research approach (Guba & Lincoln, 1989) calling for field research attempting to describe the educational setting through the participants’ eyes, and to use these eyes throughout the analytic and interpretive process.• The second assumption emerges from the theory of discursive psychology, which tries to integrate cognitive psychology with socio- cultural and anthropological theories (Pontecorvo, 1993), while emphasizing the central role that communicational processes have in learning and thinking (Edwards, 1997; Bruner, 1990).3.2 Data Collection and Discourse Analysis The focus of the research presented here was on the first phase of therecursion learning process, as it naturally evolved during 1 month of learningin 6 different cases of 11th grade classes (titled as Case 9 … Case 14; Pavel’sclass introduced earlier is titled Case 10). These classes had just begun theintermediary period of their FP course (see Lapidot et al., 1999 for details onthe course), when the learners were first exposed to the idea of recursion.The number of students varied from 7 in the smallest class (Case 9) to 22 inthe largest (Case 12). The first phase of the recursion learning process in each case beganwhen the learners participated in a three-sessions learning activity. Duringthe first session, each learner received a large sheet of paper with examplesof recursive phenomena (like in Fig. 1), and the learners jointly classifiedthese examples while working in groups of three to four learners each. In the 121
  10. 10. Vol. 12, No. 2, 2002 Collaborative Conceptual Change: The Case of Recursionsecond session, usually a week later, each group presented its classificationto the whole class, and in the third session, the teacher guided a reflectiveclass discussion toward formulating the general idea of recursion using amore formal language. The class discourse during the whole learning activity was recorded anddocumented using observational field notes, during at least four consecutivetwo-hour lessons in each of the six cases. The documentation, accordingly,detailed 50 hours of class discourse. The recordings were then fullytranscribed, and after excluding utterances not directly connected with thestudied learning activity (like two students talking about their driving lessonsafter school in Case 9), the gathered ‘raw data’ was transformed into a pull ofalmost 500 discourse episodes altogether. Whereas Pavel’s utterance is anexample of a one-utterance discourse episode, most episodes were in theform of two or more consecutive utterances expressed by two or moredifferent learners speaking about the same specific issue or theme (see theexample at the beginning of the next section). The transcriptions of these discourse episodes, together with field notes,served as the source for an inductive discourse analysis. According to thismethod of analysis, “As you read through your data, certain words, phrases,patterns of behavior, subjects’ ways of thinking, and events repeat and standout…These words and phrases are coding categories” (Bogdan & Biklen,1998, p. 171). In the first phase of analysis, three analytic perspectives, orthree different dimensions were observed in the students’ discourse: thecontent, the cognitive, and the communicative dimensions. The first contentperspective will be presented here. The last two analytic perspectives will bedealt with in a future publication. Using the content perspective, the analysis then concentrated on what thestudents talked about and used their words, phrases, drawings, and writtenproductions as coding categories. The next section presents these emergentcontent categories, interprets it as pre-conceptions, and suggests a model fororganizing these preconceptions. As it will briefly be discussed later, thesuggested model may reflect a certain kind of conceptual change that mighthave taken place in the observed classes.122
  11. 11. Dalit Levy Journal of Intelligent Systems4. RESULTS: PRECONCEPTIONS EXPRESSED BY HIGH SCHOOL STUDENTS4.1 The Interpretation of Class Episodes Before listing the key preconceptions that emerged in the course of thediscourse analysis, let us look at one class episode from Case 14, in whichHila is also talking about the tree (see Fig. 1), trying to realize what happensif you try to draw more and more levels of the tree, and the others argue withher about her realization. Hila: “Here we can’t see the end, but there is an end anyway. It clashes, these leaves will clash. They must clash.” Amos: “Theoretically you can go on.” Gil: “It can go on, but you won’t see it.” Tal: “It is repeating. Periodical. Everything here is periodical.” The episode was documented in the sixth observed class, one year afterthe documentation in Pavel’s class (Case 10). Like Pavel before, the studentshere express their ideas concerning the common features of the recursivephenomena they had just classified before. But the latter example hints atdifferent preconceptions than those expressed by Pavel’s utterance and alsohints at the students’ making use of cognitive acts like naming, comparing,classifying and generalizing (Feuerstein et al., 1980). As mentioned before,the cognitive analytic perspective will not be dealt with here. What this paperdoes focus on are the underlined words and phrases or the content expressedby each episode. Such expressions may serve as indicators, or hints, for thestudents’ unique ways of thinking about the recursive phenomena they hadbeen investigating. When Tal said, “Everything here is periodical,” sheexpressed her own way of perceiving and characterizing the variousrecursive phenomena she dealt with, using what might be called theperiodical preconception. When Pavel said, “This one is like the other butsmaller,” he expressed both the preconception of gradualism and thepreconception of likeness that has been later, while comparing it withepisodes and preconceptions from other cases, reformulated as self-similarity.These unique student-made phrases were part of the huge amount of data 123
  12. 12. Vol. 12, No. 2, 2002 Collaborative Conceptual Change: The Case of Recursiongathered, analyzed, and finally entitled as preconceptions representing thestudents’ different ways of thinking about recursion. Here we emphasize thatpreconceptions are definitely not ‘misconceptions’ and that the differentways of thinking are definitely not ‘wrong’. The label ‘preconceptions’ waschosen by considering both the conceptual nature of the discourse and theinitiative phase of the learning process in which the student had beeninvolved.4.2 Emergent Preconceptions When Discussing Recursive Phenomena Analyzing the discourse, a diverse collection of two dozens contentcategories came up, with each category including expressions that hint at asimilar way of talking and thinking about recursive phenomena. These contentcategories are regarded as preconceptions, and the whole categorical system isregarded as the network of preconceptions that evolved throughout the study.Table 1 presents one third of the categorical system, namely eight categoriesthat are considered key preconceptions. These key preconceptions appearedmost often in the students’ discourse and were remarkably associated with otherpreconceptions. Each key preconception in Table 1 is illustrated by an utteranceexpressing it. The representative utterances were selected among the datagathered at the different classes (titled as Case 9…Case 14). For an expandedview, the right column of the table presents the various other preconceptionsthat tended to be associated with each key pre-conception. So far, the collection of preconceptions emerging throughout the discourseanalysis has been briefly described. Recall that the term ‘preconception’denotes a specific way of looking at recursive phenomena and of describingcertain characteristics of such phenomena. For example, when Pavel’sclassmate looked at one item from the collection of recursive phenomena andsaid ‘there is a kind of a rule here’ (Case 10), his utterance was interpreted ashinting at regularity. In this case, the preconception of regularity denotes thatstudent’s specific way of looking for rules in the recursive phenomenon that hewas investigating. As stated before, preconceptions are not misconceptions;moreover, as the example of regularity shows, all emergent preconceptions canbe thought of as being closely related with recursion by hinting at different124
  13. 13. Dalit Levy Journal of Intelligent Systemscharacteristics of a variety of recursive phenomena. 125
  14. 14. Vol. 12, No. 2, 2002 Collaborative Conceptual Change: The Case of Recursion Two important findings can be summarized here: First, high schoolstudents indeed expressed a rich and complicated conceptual scheme whenthey were first exposed to recursion via the classification and discussion ofdifferent recursive phenomena. That complex network of preconceptions thatemerged throughout the study reflects an advanced framework of students’thinking about recursion even at that initiative phase. As the documentedlearning activity took place mainly as a collaborative discussion involving allstudents in the class, we claim that the conceptual advancement went hand inhand with the need to negotiate one’s conceptions with the others. Within theframework of social constructivism (Confrey, 1995), such a finding mightindicate that social interaction can stimulate the elaboration of conceptualknowledge (Van Boxtel et al., 2000) or, as has been previously claimed,social interaction might motivate conceptual change. The next section willfurther elaborate on that first finding. The second finding refers to some key preconceptions like Infinite orFinite, Periodical, and Gradualism (see Table 1) that were highly linked toothers, whereas other preconceptions tended to be more isolated. The moreisolated preconceptions are not referred to as ‘key preconceptions’ andtherefore are not presented in Table 1, but one can find them listed in Table 2(the non-bolded preconceptions). For example, gradualism was apparent inPavel’s utterance, as well as in many other discourse episodes as a kind ofinherent characteristic of recursive phenomena, which was always jointlyexpressed with one or more other preconceptions. Another example of ahighly linked system of preconceptions can be found in several ‘potentiallyrich episodes’, in which five or more different preconceptions wereexpressed in the same discourse episode. As has been extensively dealt withelsewhere (Levy, 2001), because of their argumentative nature, thoseepisodes had the most promising potential for conceptual change. On theother hand, the preconception Containing can also be thought of as aninherent characteristic of recursive phenomena, but when that kind ofpreconception emerged from the investigated discourse episodes—and therewere episodes of talking about containment all along the different phases ofthe learning activity—Containing tended to ‘stand by itself’, not very muchlinked to other preconceptions. The meaning of this second finding is that the126
  15. 15. Dalit Levy Journal of Intelligent Systemsstudents’ discourse not only hinted at the components of their conceptualscheme. The discourse analysis also hinted at the process of reconstructingsuch schemes by expressing linkages and relations (Hiebert & Lefevre,1986), as well as by differentiating the ‘stand alone’ components of theconceptual scheme in regard to recursion.4.3 A Suggested Model of Conceptual Change As a further analytic step, the different phases of the learning activity wereconsidered, and the preconceptions were organized according to the phases inwhich they appeared. In Phase 1, the class was first exposed to recursivephenomena, whereas Phase 2 was the classification phase. The students workedin groups of three or four each, and in Phase 3 each group presented itsclassification, categories, and criteria before the other groups. In Phase 4, theteacher guided a reflective whole class discussion. Those four phases lastedbetween two and four consecutive sessions of two-hour lessons in each case. InPhase 2, when up to six different groups of learners worked in parallel, onlyone group was observed and recorded thoroughly in each case. In addition tothat close documentation of the classification phase, the written proposedclassifi-cations of the other groups were also collected. As each group ofstudents had to offer a title for each class of recursive phenomena, and becausethe written expressions, like the documented verbal expressions, often reflectedcommon features or characteristics of a class of phenomena, the collected titleswere integral part of the data gathered at Phase 2. Table 2 shows the suggested model for organizing the preconceptions bygrey-lightning the phases that were relevant for each preconception. The modelincludes most recognized preconceptions. Two main findings were summarizedin section 4.2 above: the diversity of the preconceptions that high schoolstudents expressed, and the conceptual network that they weaved byexpressing linkages and relations among the components of their complicatedconceptual scheme. The different styles of boxes that bound three sections ofthe model (numbered 1, 2, 3 in Table 2) hint at three additional findings:1. The consistency of preconceptions: some preconceptions appeared as early as the exposure Phase 1, and continued to be expressed all along the learning activity. The most consistent-along-the-phases were the pre- 127
  16. 16. Vol. 12, No. 2, 2002 Collaborative Conceptual Change: The Case of Recursion conceptions of Infinite or Finite, Circularity, and Containing. TABLE 2 A model for organizing the preconceptions Phase Phase Phase Phase Preconception 1 1 2 3 4 Returning Infinite or Finite 1 2 Circularity Containing Split Reflection Symmetry Sophistry Self-reference Self-similarity Regularity Regular gradual recurrence Gradualism Periodical Sequential Withdrawal Infinite gradual recurrence 3 Dependency Fractal Mutuality Function that calls itself1 The grey-lighted areas indicate the relevant phases for each preconception. The boldpreconceptions are the key preconceptions.2. The cognitive potential of group classification and discussion: the group phases (Phase 2, Phase 3) motivated a rich expression of preconceptions as well as the opportunity for conceptual change. Many of the various preconceptions were rooted in these learning-without-128
  17. 17. Dalit Levy Journal of Intelligent Systems guidance phases. Following Krummheuer (1995) and others, one might suggest that the argumentative nature of the group discussions is responsible for that richness.3. The creation and refinement of a class genre appropriate for discussing the idea of recursion: a terminological shift toward and throughout the last reflective Phase 4 seemed to occur. In that phase, the students used a slightly more formal language, e.g. their use of Symmetry, Dependency, Fractal, and Mutuality. The lingual change might also reflect a conceptual change by expressing the process of collaborative reconstruction of ideas and by pointing at the communal dimension of learning (Confrey, 1995; Cobb, 1996). In that sense, the suggested model may reflect a collaborative kind of conceptual change that occurred in the observed classes. Together with the findings presented earlier, five different results havebeen discussed here. Such discussion can illuminate the process by whichlearners construct an abstract concept like recursion and can draw an interestingand unique picture concerning the ways in which students relate to thisinterdisciplinary concept, the particular learners’ language concerning recursivephenomena and the nature of the conceptual change that might take placethroughout the class discourse—a collaborative conceptual change. When anabstract and interdisciplinary concept like recursion is constructed, theconceptual change is initiated and motivated by taking part in a collaborativeand discursive learning environment. Using somewhat metaphoric language, ifwe (as constructivists) think about concepts as located in one’s mind, we mightthink about change as located somewhere in the contextual/communicative/discursive space (Vygotsky, 1978; Roth, 1999).5. IMPLICATIONS FOR UNDERSTANDING HIGH SCHOOL STUDENTS’ CONCEPTIONS AND CONCEPTUAL CHANGE In analyzing student discourse while engaged in a recursion classificationand generalization activity, one can locate some conceptual processes as theyevolve and become expressed in the natural setting, the real classroom. One 129
  18. 18. Vol. 12, No. 2, 2002 Collaborative Conceptual Change: The Case of Recursionimplication for designing learning environments for conceptual change isobvious because the studied learning activity supports students’ constructionprocesses by enabling them to engage actively and reflectively in the learningtask, either on the group discourse level or on the whole class discourse level.This implication may well be extended to other concepts via a similar groupactivity. In a sense, the studied context of learning recursion can be thought ofas an instructional context that promotes the process of conceptual change(Mason, 2001) in computer-science classes. At the same time, the studiedcontext can be used as an example of a ‘communicational space’ in a call forinvestigating other communicational spaces to better understand thecollaborative nature of conceptual change. Furthermore, the implications of the findings mentioned above couldhold both for understanding how students construct the conceptual scheme ofrecursion and for understanding more general construction andreconstruction processes. This paper points out only one such implication,which is well documented by researchers in the discipline of mathematicseducation. The preconceptions emerged in the research hint at the interestingdistinction between the more operational kind of conception and the morestructural kind of conception. This issue has been raised both by thediscourse analysis and by contemporary theories of mathematics education.Following Piaget (1980), some researchers offer to look at the process ofconstructing abstract mathematical concepts as a gradual process, in whichthe learner moves from an operational conception towards the moredeveloped structural conception (Sfard & Linchevski, 1994; Breidenbach etal., 1992). When holding an operational conception, the learners focus atactions and processes, as can be the case for the students who expresspreconceptions like Infinite or Finite, Gradualism, and Periodical. On theother hand, focusing and expressing the preconceptions of Containing,Fractal, and Self-reference, could be interpreted as representing a morestructural conception of recursion. Discovering that all the different kinds ofconceptions were jointly present in the same class was interesting. Moreover,they often harmonically existed within a single utterance expressed by thesame student, as happens in Pavel’s utterance. Such harmony contradicts130
  19. 19. Dalit Levy Journal of Intelligent Systemsformer findings concerning the superiority of the operational conception ofrecursion, even when the students expressing that kind of conception werenot novices (Aharoni, 1999). The operational conception of recursion mightbe a consequence of a program-ming-oriented thinking, constructed byoveremphasizing computing and algorithmic aspects of recursion throughouta programming-oriented curriculum. The implication for teaching computer-science concepts in general—and for teaching recursion especially—isobvious: the learners should be exposed to a broader view, to various ways ofthinking about the basic concepts of computer science, and to a larger varietyof programming as well as non-programming learning activities. In an even more general sense, we emphasize that the recognition of therole of the class discourse in the process of constructing scientific concepts“has been one of the most important conditions in making possible changesin teaching practice” (Mortimer & Machado, 2000, p. 440). Within the youngand growing research community of computer-science education, suchrecognition is a must.ACKNOWLEDGEMENTS I acknowledge the valuable contributions made by my supervisor,Professor Uri Leron and by my colleague, Tami Lapidot, for their advice andsupport.REFERENCESAbelson, H. and Sussman, G.S., with Sussman, J. 1996. Structure and Interpretation of Computer Programs, 2nd edition, Cambridge, Massa- chusetts, USA, MIT press.Aharoni, D. 1999. Undergraduate Students’ Perception of Data Structures, Unpublished Ph.D. thesis [in Hebrew]. Technion, Israel Institute of Technology.Anderson, J.R., Pirolli, P., and Farrel R. 1988. Learning to program recursive 131
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