Collaboratve Conceptual Change : The Case of Recursion

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: dality@tx.technion.ac.il

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Vol. 12, No. 2, 2002 Collaborative Conceptual Change:

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.

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Dalit Levy Journal of Intelligent Systems

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

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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.

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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

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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

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& 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

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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.

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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

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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.

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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

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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

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characteristics of a variety of recursive phenomena.

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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

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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-

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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 itself
1
The 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-

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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

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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

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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.

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