Collaboratve Conceptual Change : The Case of Recursion

113

Collaborative Conceptual Change: The Case of Recursion

Dalit Levy

Department of Education in Science and Technology, Technion–Israel Institute of Technology, Haifa 32000 Israel

ABSTRACT

The growing body of research within computer-science education has not yet focused on studying conceptual change, in comparison with the intensive account of conceptual change in other science domains. For example, although the difficulties in learning and teaching the significant concept of recursion are often referred to, the research literature barely addresses the unique ways in which students relate to this interdisciplinary concept, the particular learners’ language concerning recursive phenomena, and the processes of conceptual change. In order to fill the gap, the study here presented describes a naturalistic study in six Israeli computer-science classes and deals with a variety of conceptions emerged from analyzing the students’ discourse of recursive phenomena. The paper also suggests a model for organizing the conceptions in a way that might represent a collaborative process of conceptual change in regard to recursion in particular, and might also hint at the communal nature of conceptual change in general. In terms of this paper, that nature is referred to as ‘collaborative conceptual change’.

KEYWORDS

class discourse, computer-science education, functional programming, constructivism, qualitative research, recursion _____________________________________________________________

Reprint requests to: Dalit Levy, Department of Education in Science and Technology, Technion–Israel Institute of Technology, Haifa 32000 Israel; e-mail: [email protected]

Vol. 12, No. 2, 2002 Collaborative Conceptual Change: The Case of Recursion

114

1. INTRODUCTION

‘Look, this one is like the other but smaller’ said Pavel while analyzing the tree in Fig. 1. When saying ‘this one’, he was pointing at the right-most tree and was drawing an imaginative little circle around it. When saying ‘like the other’, he drew a much larger circle around the whole tree figure. Pavel, a sixteen years old Israeli student, was trying to draw the attention of his classmates to a certain characteristic of a recursive phenomenon that he had just recognized. At that time, Pavel’s class was at the beginning of the major part of a functional programming (FP) course, the part that dealt with recursion. Recursion is an interdisciplinary concept with many implications in programming (Hofstadter, 1979), or as Harvey and Wright (1993, p. 168) describe it: “recursion is the idea of self-reference applied to computer programs”.

The FP course taken by Pavel was the third of five curricular units, which altogether constructed the computer-science enhanced curriculum in the Carmel high school, a large school in a northern Israeli town (all names are pseudonyms). Pavel’s class—eleventh grade—was known at the Carmel school

Fig. 1: One example of a recursive phenomenon.

Dalit Levy Journal of Intelligent Systems

115

as ‘the scientific class’, a prestigious group of fifteen girls and boys. These learners were first exposed to recursion while participating in a constructivist (Ben-Ari, 2001) and interdisciplinary learning activity. The activity took place in the spacious computer-lab of the Carmel school, but all along its two sessions of two hours each, the students did not use computers but rather sat in five groups around the tables in the middle of the room. At the beginning of the first session, each learner received a large sheet of paper with fourteen photocopied examples of recursive phenomena taken from various sources (Fig. 1 was one of them). The learners were then asked to classify the examples according to certain criteria of their group’s choice, to offer a title of their own for each class of examples, and to find additional examples of recursive phenomena. The class discourse of the first session evolved as simultaneous independent group discussions, whereas in the second session a week later, each group presented its classification and titles to the whole class. The discourse then took the form of a whole class discussion, in which students of different groups were negotiating, arguing, and expressing different 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 functions using the functional language DrScheme (Felleisen et al., 1998), but they had never before been exposed to the formalism of recursive functions or to the conceptual framework of recursion. Accordingly, the students’ first experiences with recursion were taking place throughout a learning activity in which recursion was widely viewed as an interdisciplinary concept, rooted in everyday life and experience, not merely as a programming tool or a computer-science exclusive idea. Such a collaborative and interdisciplinary learning activity could stimulate a very rich class discourse concerning both the specific examples of recursive phenomena and the general idea of recursion, when the learners express their unique way of conceiving, thinking and understanding. During one such rich class discourse, Pavel was documented expressing his own understanding. That idiosyncratic utterance, ‘this one is like the other but smaller’, has later been categorized with other similar students’ expressions, and has been interpreted as related to the conceptions of self-similarity and gradualism. These conceptions are entitled


116

in the study ‘preconceptions’, which is due to the initiative phase of the learning process in which the students were involved. But as will be discussed later, the complex network of preconceptions that emerged throughout the study reflects an advanced framework of students’ thinking about 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 a field study focusing on high school students’ discourse of recursive phenomena, while trying to trace conceptual change with regard to recursion. Note that much of the research into the learning of recursion has taken place within the framework of learning to program (for example, George, 2000; Wu et al., 1998; Segal, 1995), whereas less emphasis has been placed on understanding how learners construct the concept of recursion as a broad, abstract, and interdisciplinary concept. In addition, all concerned seem to agree on the difficult nature of recursion for novices studying computer-science, both for college and university students and for younger high school students, but the literature hardly refers the latter. Moreover, the educational research has not emphasized the learners’ voice or the learning processes, as they show up in a natural setting, in a real class dealing with computer-science concepts (Booth, 1993 is one exception). And finally, in deep contrast to the variety of publications in the field of conceptual change that one can find regarding each of the scientific disciplines being learned at high school (Duit, 2002; Soto & Sanjose, 2002), the focus on conceptual change with regard to computer-science ideas and concepts has been left aside.

The present paper tries to shed some light on these neglected aspects of 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.


117

2. THEORETICAL BACKGROUND

When studying conceptual change in regard to recursion, one should take into account both the framework within which most conceptual change studies have been conducted and the literature on learning and teaching recursion. The first kind of background can be thought of as a general background in the established discipline of science education, whereas the second is more specific and is located within the relatively new field of computer science education.

2.1 The Focus on Conceptual Change in Science Education

For almost three decades, conceptual change has been regarded as a most powerful frame for research on teaching and learning science (Duit, 2002). Throughout these years, the notion that students often enter the science classes with prior knowledge, ideas, and beliefs about the phenomena and concepts to be taught became clear. Since this prior knowledge often stands in contrast with what is regarded as ‘scientific knowledge’, the general term ‘misconceptions’ was introduced, and the learners’ misconceptions were thoroughly investigated (Wandersee et al., 1993), only to be left aside in favor of the less judgmental term ‘alternative conceptions’. In a paper criticizing the dominancy of the focus on misconceptions in the field of science education, the writers recommend to move toward tracing conceptual change 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). A constructivist belief is that knowledge is necessarily a product of our own cognitive acts and that we construct our understandings through our experiences (Confrey, 1995). In such a view, learning science means that students themselves are constructing and reorganizing their own conceptual frameworks, in a “gradual process during which initial conceptual structures


118

based on … interpretations of everyday experience are continuously enriched and restructured” (Vosniadou & Ioannides, 1998, p. 1213). The constructivist science teacher should encourage a reflective discussion to expose the learners to new conceptions and to different ideas offered by others, in addition to her or his responsibility for offering generalizations and formal terminology. In other words, the teacher’s role is to navigate the discussion toward creating a ‘taken-as-shared’ meaning for the scientific concepts being learned in the class (Cobb et al., 1992) and thus allowing some conceptual change to occur.

In this paper, conceptual change denotes pathways from students’ existing conceptual frameworks to the computer science concept to be learned, which in our case is the concept of recursion. To trace conceptual change, the researcher observed and documented students participating in a constructivist and interdisciplinary learning activity. As the next sections show, the protocols of that activity shed light both on the private conceptual frameworks students carried with them when entering the class and on a specific kind of conceptual change which will be later described as a ‘collaborative conceptual change’. A main claim of this paper is that constructivist learning activities like the one investigated (named CGA, see Levy, 2001) can be characterized by a collaborative conceptual change. In other words, although the changes occur in one’s mind and upon one’s conceptual framework, and although each student individually constructs her or his framework, conceptual change is often (always?) motivated by taking part in more societal and communicational learning activities. In more general terms, this paper (like Duit, 2002 and others) calls for merging sociocultural 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 most interesting here. Hofstadter’s (1979) book Godel, Escher, Bach: An Eternal Golden Braid gives a comprehensive account. Other books mainly deal with the programming aspects of recursion (for example, Roberts, 1986; Abelson


119

& Sussman, 1996), as does most of the educational literature on recursion. Issues of learning recursion sometimes appear in textbooks together with a warning 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 the difficulty. 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. Many

writers tend to agree upon one recommendation: “In order to develop a more complete understanding of the topic, it is important for the student to examine recursion from several different perspectives” (Roberts, 1986, p. vii ). In this spirit, Ben-Ari (1997) describes a teaching approach that strongly couples dramatizations of simple, real-world problems with analogous recursive programs; Harvey offers several explanations of recursive programs using different models (Harvey, 1985; Harvey & Wright, 1993); Astrachan (1994) declares that students should be shown as many examples as possible for them to come to “believe” in recursion; and George (2000), as well as Bhuiyan et al. (1994), develop a computerized environment designed


120

to help students create different methods of generating recursive programs. All concerned seem to agree on the difficulty of learning and teaching

recursion. Some recommendations for teaching have been made. But when trying to find “how it is” in a real class, in a natural educational setting, the research literature in computer science education is less helpful. Among the few naturalistic accounts, one can find Booth’s phenomenographic studies focusing on students’ conceptions of programming, including their conceptions of recursion (Booth, 1993), and some interview-based studies examining beliefs, misconceptions, and programming errors of students learning recursion (Segal, 1995; Lee & Lehrer, 1987). Still, the question of what a computer science class looks like when it deals with recursion remains unanswered.

As has been mentioned earlier, the gap is most apparent when looking at computer-science learning through a broader lens. When observing an inter-disciplinary and constructivist learning environment, one could also ask what language do the teacher and the students use? How can one characterize the class discourse? How does this discourse reveal the formal aspects of recursion and in what ways does it expose the difficulties? What change does the class discourse reflects, and is it a conceptual change? And what are the affective, social and communicational aspects involved in the conceptual change regarding recursion? These questions have guided a naturalistic research on learning recursion in Israeli high schools, conducted as part of the author’s graduate studies during the years 1998 to 2001.

In this paper, only one part of the overall research is discussed. In this part, the focus was on the first phase of the learning process and the questions 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, the

answers to theses question are widely discussed.


121

3. METHOD

3.1 The Research Goal

As briefly stated in the introduction, the research goal was to document and analyze learners’ discourse of recursive phenomena, as a way to look at recursion through the eyes of the learners and to help in understanding the learners’ unique ways of speaking and thinking about the general idea of recursion. The formulation of such a research goal reflects two methodological 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 the recursion learning process, as it naturally evolved during 1 month of learning in 6 different cases of 11th grade classes (titled as Case 9 … Case 14; Pavel’s class introduced earlier is titled Case 10). These classes had just begun the intermediary period of their FP course (see Lapidot et al., 1999 for details on the 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 in the largest (Case 12).

The first phase of the recursion learning process in each case began when the learners participated in a three-sessions learning activity. During the first session, each learner received a large sheet of paper with examples of recursive phenomena (like in Fig. 1), and the learners jointly classified these examples while working in groups of three to four learners each. In the


122

second session, usually a week later, each group presented its classification to the whole class, and in the third session, the teacher guided a reflective class discussion toward formulating the general idea of recursion using a more formal language.

The class discourse during the whole learning activity was recorded and documented using observational field notes, during at least four consecutive two-hour lessons in each of the six cases. The documentation, accordingly, detailed 50 hours of class discourse. The recordings were then fully transcribed, and after excluding utterances not directly connected with the studied learning activity (like two students talking about their driving lessons after school in Case 9), the gathered ‘raw data’ was transformed into a pull of almost 500 discourse episodes altogether. Whereas Pavel’s utterance is an example of a one-utterance discourse episode, most episodes were in the form of two or more consecutive utterances expressed by two or more different learners speaking about the same specific issue or theme (see the example 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 this method of analysis, “As you read through your data, certain words, phrases, patterns of behavior, subjects’ ways of thinking, and events repeat and stand out…These words and phrases are coding categories” (Bogdan & Biklen, 1998, p. 171). In the first phase of analysis, three analytic perspectives, or three different dimensions were observed in the students’ discourse: the content, the cognitive, and the communicative dimensions. The first content perspective will be presented here. The last two analytic perspectives will be dealt with in a future publication.

Using the content perspective, the analysis then concentrated on what the students talked about and used their words, phrases, drawings, and written productions as coding categories. The next section presents these emergent content categories, interprets it as pre-conceptions, and suggests a model for organizing these preconceptions. As it will briefly be discussed later, the suggested model may reflect a certain kind of conceptual change that might have taken place in the observed classes.


123

4. RESULTS: PRECONCEPTIONS EXPRESSED BY HIGH SCHOOL

STUDENTS

4.1 The Interpretation of Class Episodes

Before listing the key preconceptions that emerged in the course of the discourse analysis, let us look at one class episode from Case 14, in which Hila is also talking about the tree (see Fig. 1), trying to realize what happens if you try to draw more and more levels of the tree, and the others argue with her 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 after

the documentation in Pavel’s class (Case 10). Like Pavel before, the students here express their ideas concerning the common features of the recursive phenomena they had just classified before. But the latter example hints at different preconceptions than those expressed by Pavel’s utterance and also hints 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 paper does focus on are the underlined words and phrases or the content expressed by each episode. Such expressions may serve as indicators, or hints, for the students’ unique ways of thinking about the recursive phenomena they had been investigating. When Tal said, “Everything here is periodical,” she expressed her own way of perceiving and characterizing the various recursive phenomena she dealt with, using what might be called the periodical preconception. When Pavel said, “This one is like the other but smaller,” he expressed both the preconception of gradualism and the preconception of likeness that has been later, while comparing it with episodes and preconceptions from other cases, reformulated as self-similarity. These unique student-made phrases were part of the huge amount of data


124

gathered, analyzed, and finally entitled as preconceptions representing the students’ different ways of thinking about recursion. Here we emphasize that preconceptions are definitely not ‘misconceptions’ and that the different ways of thinking are definitely not ‘wrong’. The label ‘preconceptions’ was chosen by considering both the conceptual nature of the discourse and the initiative phase of the learning process in which the student had been involved.

4.2 Emergent Preconceptions When Discussing Recursive Phenomena

Analyzing the discourse, a diverse collection of two dozens content categories came up, with each category including expressions that hint at a similar way of talking and thinking about recursive phenomena. These content categories are regarded as preconceptions, and the whole categorical system is regarded as the network of preconceptions that evolved throughout the study. Table 1 presents one third of the categorical system, namely eight categories that are considered key preconceptions. These key preconceptions appeared most often in the students’ discourse and were remarkably associated with other preconceptions. Each key preconception in Table 1 is illustrated by an utterance expressing it. The representative utterances were selected among the data gathered at the different classes (titled as Case 9…Case 14). For an expanded view, the right column of the table presents the various other preconceptions that tended to be associated with each key pre-conception.

So far, the collection of preconceptions emerging throughout the discourse analysis has been briefly described. Recall that the term ‘preconception’ denotes a specific way of looking at recursive phenomena and of describing certain characteristics of such phenomena. For example, when Pavel’s classmate looked at one item from the collection of recursive phenomena and said ‘there is a kind of a rule here’ (Case 10), his utterance was interpreted as hinting at regularity. In this case, the preconception of regularity denotes that student’s specific way of looking for rules in the recursive phenomenon that he was investigating. As stated before, preconceptions are not misconceptions; moreover, as the example of regularity shows, all emergent preconceptions can be thought of as being closely related with recursion by hinting at different


125

characteristics of a variety of recursive phenomena.


126

Two important findings can be summarized here: First, high school students indeed expressed a rich and complicated conceptual scheme when they were first exposed to recursion via the classification and discussion of different recursive phenomena. That complex network of preconceptions that emerged throughout the study reflects an advanced framework of students’ thinking about recursion even at that initiative phase. As the documented learning activity took place mainly as a collaborative discussion involving all students in the class, we claim that the conceptual advancement went hand in hand with the need to negotiate one’s conceptions with the others. Within the framework of social constructivism (Confrey, 1995), such a finding might indicate that social interaction can stimulate the elaboration of conceptual knowledge (Van Boxtel et al., 2000) or, as has been previously claimed, social interaction might motivate conceptual change. The next section will further elaborate on that first finding.

The second finding refers to some key preconceptions like Infinite or Finite, Periodical, and Gradualism (see Table 1) that were highly linked to others, whereas other preconceptions tended to be more isolated. The more isolated preconceptions are not referred to as ‘key preconceptions’ and therefore are not presented in Table 1, but one can find them listed in Table 2 (the non-bolded preconceptions). For example, gradualism was apparent in Pavel’s utterance, as well as in many other discourse episodes as a kind of inherent characteristic of recursive phenomena, which was always jointly expressed with one or more other preconceptions. Another example of a highly linked system of preconceptions can be found in several ‘potentially rich episodes’, in which five or more different preconceptions were expressed in the same discourse episode. As has been extensively dealt with elsewhere (Levy, 2001), because of their argumentative nature, those episodes had the most promising potential for conceptual change. On the other hand, the preconception Containing can also be thought of as an inherent characteristic of recursive phenomena, but when that kind of preconception emerged from the investigated discourse episodes—and there were episodes of talking about containment all along the different phases of the learning activity—Containing tended to ‘stand by itself’, not very much linked to other preconceptions. The meaning of this second finding is that the


127

students’ discourse not only hinted at the components of their conceptual scheme. The discourse analysis also hinted at the process of reconstructing such schemes by expressing linkages and relations (Hiebert & Lefevre, 1986), as well as by differentiating the ‘stand alone’ components of the conceptual 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 were considered, and the preconceptions were organized according to the phases in which they appeared. In Phase 1, the class was first exposed to recursive phenomena, whereas Phase 2 was the classification phase. The students worked in groups of three or four each, and in Phase 3 each group presented its classification, categories, and criteria before the other groups. In Phase 4, the teacher guided a reflective whole class discussion. Those four phases lasted between two and four consecutive sessions of two-hour lessons in each case. In Phase 2, when up to six different groups of learners worked in parallel, only one group was observed and recorded thoroughly in each case. In addition to that close documentation of the classification phase, the written proposed classifi-cations of the other groups were also collected. As each group of students had to offer a title for each class of recursive phenomena, and because the written expressions, like the documented verbal expressions, often reflected common features or characteristics of a class of phenomena, the collected titles were integral part of the data gathered at Phase 2.

Table 2 shows the suggested model for organizing the preconceptions by grey-lightning the phases that were relevant for each preconception. The model includes most recognized preconceptions. Two main findings were summarized in section 4.2 above: the diversity of the preconceptions that high school students expressed, and the conceptual network that they weaved by expressing linkages and relations among the components of their complicated conceptual scheme. The different styles of boxes that bound three sections of the 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-


128

conceptions of Infinite or Finite, Circularity, and Containing. TABLE 2

A model for organizing the preconceptions

Preconception Phase

11

Phase

2

Phase

3

Phase

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 itself

1The grey-lighted areas indicate the relevant phases for each preconception. The bold preconceptions 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-


129

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 have

been discussed here. Such discussion can illuminate the process by which learners construct an abstract concept like recursion and can draw an interesting and unique picture concerning the ways in which students relate to this interdisciplinary concept, the particular learners’ language concerning recursive phenomena and the nature of the conceptual change that might take place throughout the class discourse—a collaborative conceptual change. When an abstract and interdisciplinary concept like recursion is constructed, the conceptual change is initiated and motivated by taking part in a collaborative and discursive learning environment. Using somewhat metaphoric language, if we (as constructivists) think about concepts as located in one’s mind, we might think 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 classification and generalization activity, one can locate some conceptual processes as they evolve and become expressed in the natural setting, the real classroom. One


130

implication for designing learning environments for conceptual change is obvious because the studied learning activity supports students’ construction processes by enabling them to engage actively and reflectively in the learning task, 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 group activity. In a sense, the studied context of learning recursion can be thought of as an instructional context that promotes the process of conceptual change (Mason, 2001) in computer-science classes. At the same time, the studied context can be used as an example of a ‘communicational space’ in a call for investigating other communicational spaces to better understand the collaborative nature of conceptual change.

Furthermore, the implications of the findings mentioned above could hold both for understanding how students construct the conceptual scheme of recursion and for understanding more general construction and reconstruction processes. This paper points out only one such implication, which is well documented by researchers in the discipline of mathematics education.

The preconceptions emerged in the research hint at the interesting distinction between the more operational kind of conception and the more structural kind of conception. This issue has been raised both by the discourse analysis and by contemporary theories of mathematics education. Following Piaget (1980), some researchers offer to look at the process of constructing abstract mathematical concepts as a gradual process, in which the learner moves from an operational conception towards the more developed structural conception (Sfard & Linchevski, 1994; Breidenbach et al., 1992). When holding an operational conception, the learners focus at actions and processes, as can be the case for the students who express preconceptions like Infinite or Finite, Gradualism, and Periodical. On the other hand, focusing and expressing the preconceptions of Containing, Fractal, and Self-reference, could be interpreted as representing a more structural conception of recursion. Discovering that all the different kinds of conceptions were jointly present in the same class was interesting. Moreover, they often harmonically existed within a single utterance expressed by the same student, as happens in Pavel’s utterance. Such harmony contradicts


131

former findings concerning the superiority of the operational conception of recursion, even when the students expressing that kind of conception were not novices (Aharoni, 1999). The operational conception of recursion might be a consequence of a program-ming-oriented thinking, constructed by overemphasizing computing and algorithmic aspects of recursion throughout a programming-oriented curriculum. The implication for teaching computer-science concepts in general—and for teaching recursion especially—is obvious: the learners should be exposed to a broader view, to various ways of thinking about the basic concepts of computer science, and to a larger variety of programming as well as non-programming learning activities.

In an even more general sense, we emphasize that the recognition of the role of the class discourse in the process of constructing scientific concepts “has been one of the most important conditions in making possible changes in teaching practice” (Mortimer & Machado, 2000, p. 440). Within the young and growing research community of computer-science education, such recognition 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 and support.

REFERENCES

Abelson, 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


132

functions, in: The Nature of Expertise, edited by Chi, M.T., Glaser, R., and Farr M.J., Hillsdale, New Jersey, USA, Lawrence Erlbaum Associates, 153–183.

Astrachan, O. 1994. Self-reference is an illustrative essential, Proceedings of the 25st SIGCSE technical symposium on computer science education, Phoenix, Arizona, USA, 238-242.

Ben-Ari, M. 2001. Constructivism in computer science education, Journal of Computers in Mathematics and Science Teaching, 201, 45–73.

Ben-Ari, M. 1997. Recursion: from drama to program. Journal of Computer Science Education, 113, 9–12.

Bhuiyan, S., Greer, J.E., and McCalla, G.I. 1994. Supporting the learning of recursive problem solving. Interactive Learning Environments, 42, 115–139.

Bogdan, R.C., and Biklen, S.K. 1998. Qualitative Research for Education: an Introduction to Theory and Methods, 3rd edition, Boston, Massachusetts, USA, Allyn & Bacon.

Booth, S. 1993. The structure of programming awareness and learning to program, 5th Conference of European Association for Research in Learning and Instruction EARLI, Aix-en-Provence.

Breidenbach, D., Dubinsky, E., Hawks, J., and Nichols, D. 1992. Develop-mental of the process conception of function, Educational Studies in Mathematics 23, 247–285.

Bruner, J. 1990. The proper study of man, Acts of Meaning (Jerusalem-Harvard Lectures), Chapter 1, Cambridge, Mass, Harvard University press, 1–32.

Cobb P. 1996. Accounting for mathematical learning in the social context of the classroom, Eighth International Congress on Mathematics Education ICME 8, Seville, Spain.

Cobb P., Yackel, E., and Wood, T. 1992. Interaction and learning in mathematics classroom situations, Educational Studies in Mathematics Education, 23, 99–122.

Confrey J. 1995. How compatible are radical constructivism, sociocultural approaches, and social constructivism? In: Constructivism in Education, edited by Steffe, L.P., and Gale, J., Hillsdale, New Jersey, USA, Lawrence


133

Erlbaum Associates, 185–225. Duit, R. 2002. Conceptual change—still a powerful frame for improving

science teching and learning? Proceedings of the Third European Symposium on Conceptual Change, Turku, Finland, 5–16.

Edwards, D. 1997. Discourse and Cognition, London, UK: SAGE. Feuerstein, R., Rand, Y., Hoffman, M.B., and Miller, R. 1980. Instrumental

Enrichment—An Intervention Program for Cognitive Modifiability, Baltimore, Maryland, USA, University Park Press.

George, C.E. 2000. EROSI—Visualizing recursion and discovering new errors, Proceedings of the 31st SIGCSE Technical Symposium on Computer-science Education, Austin, Texas, USA, 305–309.

Guba, E.G., and Lincoln, Y.S. 1989. Fourth Generation Evaluation, London, UK, Sage Publications.

Felleisen, M., Findler, R.B., Flatt, M., and Krishnamurti, S. 1998. How to Design Programs, Houston, Texas, USA, Rice University.

Harvey, B., and Wright, M. 1993. Simply Scheme—Introducing Computer-Science, Cambridge, Massachusetts, USA, MIT press.

Harvey, B. 1985. Computer Science Logo Style, Volume 1: Intermediate Programming, Cambridge, Massachusetts, USA, MIT press.

Hiebert, J., and Lefevre, P. 1986. Conceptual and procedural knowledge in mathematics: an introductory analysis, In: Conceptual and Procedural Knowledge: The Case of Mathematics, edited by J. Hiebert, Hillsdale, New Jersey, USA: Lawrence Erlbaum, 1–27.

Hofstadter, D. 1979. Godel, Escher, Bach: An Eternal Golden Braid, New York, NY, USA, Basic Books.

Krummheuer, G. 1995. The Ethnography of Argumentation, in: The Emergence of Mathematical Meaning–Interactions in Classroom Culture, edited by Cobb, P., and H. Beauersfeld, H., Hilsdale, New Jersey, USA: Erlbaum. pp. 229–269.

Kurland, D.M. and Pea, R.D. 1983. Children’s mental models of recursive LOGO programs, Proceedings of the 5th Annual Conference of the Cognitive Science Society, Session 4, 1–5.

Lapidot, T., Levy, D., and Paz, T. 1999. Implementing constructivist ideas in a functional programming course for secondary school, Workshop on


134

Functional and Declarative Programming in Education, Paris, France. Lee, O., and Lehrer, R. 1987. Conjectures concerning the origins of mis-

conceptions in LOGO, Annual Meeting of the American Educational Research Association AERA, Washington, DC, USA.

Leron, U. 1988. What makes recursion hard, Sixth International Congress on Mathematics Education ICME6, Budapest, Hungary.

Levy, D. 2001. Insights and conflicts in discussing recursion: A case study, Computer-Science EducationI, 114, 305–322.

Mason, L. 2001. Introduction: Instructional practices for conceptual change in science domains, Learning and Instruction, 114, 5, 259–263.

McCalla, G.I. and Greer, J.E. 1993. Two and one-half approaches to helping novices learn recursion, in: Cognitive Models and Intelligent Environments for Learning Programming, edited by Lemout, E., Du Boulay, B., and Dettori G., New York, NY, USA, Springer Verlag, 185–197.

Mortimer, F.M. and Machado, A.H. 2000. Anomalies and conflicts in class-room discourse, Science Education, 84, 429–444.

Pirolli, P.L., and Anderson, J.R. 1985. The role of learning from examples in the acquisition of recursive programming skills, Canadian Journal of Psychology, 39, 240–272.

Piaget, J. 1980. Adaptation and Intelligence: Organic Selection and Pheno-copy, 3rd edition, Chicago, Illinois, USA, University of Chicago Press.

Pontecorvo, C. 1993. Forms of discourse and shared thinking, Cognition and Instruction, 113, 4, 189–196.

Roberts, E.S. 1986. Thinking Recursively, New York, NY, USA, John Wiley & Sons.

Roth, W.M. 1999. Discourse and agency in school science laboratories, Discourse Processes, 28, 27–60.

Segal, J. 1995. Empirical studies of functional programming learners evaluating recursive functions, Instructional Sciences, 22, 385–411.

Sfard, A., and Linchevski, L. 1994. The gains and pitfalls of reification—the case of algebra, Educational Studies in Mathematics, 26, 191–228.

Soto, C., and Sanjose, V. 2002. Research on conceptual change: a review in science education, Third European Symposium on Conceptual Change, Turku, Finland.


135

Smith, J.P., diSessa, A.A., and Roschelle, J. 1993. Misconceptions reconceived: a Constructivist analysis of knowledge in transition, The Journal of the Learning Sciences, 32, 115–163.

Troy, M.E., and Early, G. 1992. Unraveling recursion, Part I and II. The Computing Teacher, March–April.

Van Boxtel, C., Van der Linden, J., and Kanselaar, G. 2000. Collaborative learning tasks and the elaboration of conceptual knowledge, Learning and Instruction, 104, 311–330.

Vygotsky, L.S. 1978. Mind in Society: The Development of Higher Psychological Processes, Cambridge, Massachusetts, USA: Harvard University Press.

Vosniadou, S., and Ioannides, C. 1998. From conceptual change to science education: a psychological point of view, International Journal of Science Education, 20, 1213–1230.

Wu, C., Dale, N.B., and Bethel, L.J. 1998. Conceptual models and cognitive learning styles in teaching recursion, SIGCSE Bulletin, 292–296.