The Curriculum Journal, Vol. 15, No. 3, Autumn 2004
ICT resources in the teaching of mathematics: between computer and school technologies. A case-study Alejandra Bosco* Universitat Autonoma de Barcelona
This article is based on a research project called ‘Information and communications technology (ICT) resources in school organizational and symbolic technology: a case-study’ which was carried out in a primary school in Barcelona, Spain. The research looked at various interactions that took place between the computer as a teaching tool, the teachers’ teaching practices and the children’s learning. The observations of these interactions focus on both the process of integration of the computer in school and the way in which this medium affects and is affected by school teaching practices, and the learning process of the children partaking in those practices.
Keywords: ICT; teaching practices; educational interaction; learning strategies; innovation; mathematics teaching Introduction Our working hypothesis considered that ICT takes on new meanings when it becomes a teaching and learning tool for teachers and students. Our concern was related to the analysis and interpretation of the various interactions that took place around the computer as a teaching tool, specifically, the teachers’ practices and the students’ learning within a given school. During the school year of 1997–8 we conducted a case-study in a primary school that we considered typical as regards the adoption of ICT. We focused on the teaching of mathematics to classes of 10 and 11 year olds, where computers were most widely used. In the first part of the article we explain the research problem and its theoretical background, as well as the methodology used. The second part presents our findings and concluding remarks. Sharing the results of our study could be of use not only to the people directly involved in the use of ICT resources in schools, but also to educational policy-makers, who are often not informed of the processes as they are actually experienced in schools. *Corresponding author: Email:
[email protected] ISSN 0958-5176 (print)/ISSN 1469-3704 (online)/04/030265–16 # 2004 British Curriculum Foundation DOI: 10.1080/09585170412331311510
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Theoretical representation of the problem There are two main issues addressed in this study, which can be formulated as two main sets of research questions: (1) What ICT (both hardware and software) does the school have, and why? What characteristics do the main teaching ICT programs have as mediators of the teaching–learning process? (2) How does ICT interlock with the ways of organizing and representing the teaching activities, regarding the tasks carried out in class and the interaction (a) between teacher and students; (b) among the children; and (c) between the teaching–learning agents and the computer? The computer as a symbolic and artifactual technology: a socio-cultural approach ICT is characterized as an artifactual and symbolic technology (Alvarez Revilla et al., 1993) that is brought into a given school culture and context and in some teaching practices informed by knowledge and professional practices in order to improve the students’ learning. As an artifact, it is a material tool consisting of an identifiable unit, made up of material components, which occupies a space and possesses a certain independence from human agents in order to develop its activity (Alvarez Revilla et al., 1993, p. 47). This artifact is a potent tool for the automatic treatment of symbolic information. In this study, together with its artifactual dimension, we have considered the concept of logical supports as systems or symbolic tools. Their language promotes a certain type of representation of the information for the user, as well as a way to interact with it. Specifically, a symbolic technology is one through which: representation or construction techniques reproduce a state of affairs, substituting the real components with signs or, from these . . . describe properties and relations among the construction of signs. (Alvarez Revilla et al., 1993, p. 29)
These representations are related to Vygotsky’s view (1979) of the role of symbolic tools, such as language, in the development of the higher processes of thought, and therefore their importance for learning. When the signs and symbols of a culture are internalized through processes of a psychological nature, there begins a new phase of development. The elementary cognitive processes undergo a radical transformation under the mediation of these new instruments, generating superior cognitive processes. More recently, several authors, like Wertsch (1991), Roggoff (1993) and de Pablos et al. (1999) have pondered the consequences of mediating systems on the thinking processes of agents. According to Wertsch, when we use an instrument we are taking our mind beyond our skin. The agent of the action mediated by any instrument is an individual who acts together with the mediating instrument. Bateson (1972) presented the following illustration: suppose I am a blind man and I use a stick. I go tap, tap, tap. Where do I start? Is my mental system bounded at the handle of the stick? Does it end at my skin? Does it start halfway up the stick? Does it start at the tip of the stick? But these are nonsense questions. The stick is a pathway along which
ICT and mathematics 267 transformations of difference are being transmitted. The way to demarcate the system is to draw the limiting line in such a way that you do not cut any of these pathways, in ways which leave things inexplicable. (Wertsch, 1991, p. 50)
But mediation is not limited to a tool and its individual human agent. Indeed, mediating instruments arise in response to a huge series of social forces, besides the psychological functions that are instantiated in the example. It is therefore necessary to analyse the suppositions resulting from the influence of the socio-cultural environment in order to elucidate which teaching and learning ways they promote. In the case of the computer, both the origins of the type of software chosen and the facts that compelled that particular choice constitute cultural contexts and they have an instrumental role in the thought configuration of its users, as Cole & Scribner (1982) and Newmann et al. (1991) have pointed out. From the educational system and school culture to the teachers’ practices and knowledge: a stage for ICT teaching When the computer arrives in school, it starts to interact with different elements in that particular institution. These interactions are the products of different decisions that have been taken along a lengthy process. They encompass the educational macro-policies in a country or community, through to the ways in which the teachers organize their teaching practices, to how the students approach the different objects of knowledge with which they are presented in school. Some of those decisions take place in schools, which have a particular way of doing things, and a given school culture (Lacasa, 1994) will result in an important determinant for the existence of the computer. That is, the school culture determines the very existence of the computer, the peculiar way in which it is used, the choice of programs (logical supports) and especially the decisions on how to teach. The use of the computer, then, is modelled by school and classroom technologies which organize its actions, and by a symbolic technology, as these actions are represented in a given way. On one hand, it functions on a stage—the school. On the other hand, it is modelled by representations and symbolic exchanges that inform the meaning and guide the actions of teaching, learning, knowing, evaluating, and are therefore directly related to the way in which the curriculum is carried out, even if this is not always explicit (Mercado, 1991; Salgueiro, 1998). Teachers enact the curriculum, interpreting the whole set of elements implied in a class situation, many of them pre-established in a written document. The use of all teaching material is circumscribed by these interpretations. It is also unavoidably transformed in accordance with some knowledge, some values and some concrete learning situations. Every teaching situation, and the use of the computer along with it, is reconstructed in a process that interacts in a particular way with its tools, forming a social construct. Some authors, writing from a naturalistic perspective, agree in stating that all teachers need to see the implementation of ICT in terms of their own teaching interests and values (Olson, 1984, 1986; Martin, 1988; Trumbull, 1989;1 Alonso,
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1992; Guitert, 1995). Furthermore, other authors (Parr, 2000; Pelgrum, 2001) recommend some organizational and pedagogical conditions which schools should take into account in order to be more successful in relation to students’ learning. Aspects like the compatibility between software and teacher pedagogical approach, or different organizing actions around the computer, are among the most relevant. Methodology The case-study was carried out within an ethnographic perspective. The specific, unique situation, the particular system (Smith, 1978) that constitutes the case in this study is a primary school in Barcelona, which we thought of as ‘a typical case’ (Goetz & Lecompte, 1988). On the one hand, this school favoured the use of ICT resources. On the other hand, it integrated learning comprehensively during the whole schooling time. The study focused on the mathematics classes of 10 and 11 year olds. Both these particular agents and the school subject constituted the set of factors around which the computer was most used. The data collection followed the usual recommendations for the development of ethnographic studies: lengthy stay in the field (1997–8 school year), direct observation, interviews, and analysis of documents and artifacts. One year later, in 1999, we came back to the field several times in order to contrast and complete our data collection. Results The first of the following sections describes certain characteristics of the school and, specifically, its view on ICT and the role of the educational administration as the provider of the necessary equipment. The data deal with this first issue of what equipment, machines and software are available to the school. The second section presents an analysis of the type of learning and teaching fostered by the most used programs. The third section analyses the role of the computer in the development of the curriculum, as well as the types of relationships that the use of ICT provokes, both between teacher and students and among the children in their classes. Finally, in the fourth section, we look at the learning strategies that children use in computerassisted classes. Both the third and fourth sections respond to the second research question, that is, what happens to the ways of organizing and representing the teaching activities of the class, the relations between the teachers and the students as well as the children’s learning when they are all interacting with the computer? About the school and the new technologies The research was carried out in a primary school in Barcelona, founded in 1976 by a group of teachers who had a clear view of the pedagogical orientation they wanted the school to have. The staff still includes most of those teachers. The founding spirit and the sheer fact that most teachers had been working together for a period of ten to twenty years made for an autonomous, co-operative team, which favoured the venture
ICT and mathematics 269 of different projects geared towards school improvement. Among these projects, one that was undertaken in 1988 concerned the introduction of new technologies into their teaching practices. ICT in the school: articulation with ‘PIE’ (Program of Educational ICT). In agreement with the interest of the school to promote the frequent use of different ICT tools, the teaching staff proposed the idea of setting up an ICT classroom. In spite of successive demands to the local administration, sustained work on different projects and a struggle to develop the initiative over nearly ten years, ICT implementation did not come into being until 1993. Even so, the school is far from having cutting-edge technology, due to the fact that it depends on the administration (specifically the Program of Educational ICT—or PIE) and the low budget that the administration allocates to primary school ICT equipment (priority is given to secondary schools for ICT equipment). As for the proposal for pedagogical work with the computer, it was carried out with the support of the PIE, through the ‘Seminaris d’ Actualitzacio´ en Tecnologia de la Informacio´’ (SATIs)—seminars for continuous training that take place five to seven times a year and assign the educational programs, material resources (ICT material, programs, CD-ROMS, etc.), provide orientation on the use of the materials and the human resources (technical support, guidance, training) to back up the ICT projects. These meetings focus on the presentation of educational software and suggest activities to use the programs in a specific field of knowledge or level (beginning cycle, natural sciences, etc.). Sometimes they also deal with a monographic subject, such as the ‘internet’. From the SATI proposals, each school co-ordinator chooses the type of programs which he or she believes fits best the pedagogical line followed by his or her school, or which best meets its needs. The daily life of ICT in the school. At the time of the study, the classroom possessed seven compatible personal computers, three Pentium Multimedia II and four 486, connected to an ink jet printer. It also had a scanner. Table 1 shows the main available software support: After attending the SATI meeting, the ICT co-ordinator (a teacher who assumes responsibility for this area) is in charge of informing each teacher about the programs that have been assigned to the school, giving individual help and guidance, so the Table 1. Main programs available in the school computers Type of application
Name of application
Text processor Drawing programs Language Mathematics English Music
Write, Ami Pro, Sopa de Iletres, Microsoft Works, Meca Dpaint, Paintbrush, Paint Shop Pro, PSP Browser, Mosaic Clic and Babel Applications (exercises) Clic Applications (exercises), Winlogo, Cabrigeometra (microworlds) Clic and Babel Applications (exercises) Music and Music Time
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teachers can get to know the programs and develop their projects. Other coordination responsibilities have to do with the technical maintenance of the classroom: installation of equipment and programs, connection of accessories such as scanner, printer etc. Each class is assigned approximately one hour a week for use of the ICT classroom to work on the school subject and with the program that the teacher has decided to use. As the teacher cannot work with the whole group at the same time, because of computer logistics, the students take turns so that each half-class group uses the computer every other week, with two to three students per computer (depending on the number of people in the class). This ICT time has to coincide with the flexible time of the group. As a result of this organization, each student only works with the computer about twice a month in each subject where the teacher decides to use it. Different problems arise from this type of organization, mainly regarding the difficulty of working out a schedule for a flexible group which also has to fit in with the other schedules of the school, which, in turn, has a series of other priorities, partly due to the scarcity of human resources. The software used by the school In order to acquire the software, schools depend on their production and/or provision by the administration (PIE) or their availability in the commercial market, although the latter is hardly an option considering the reduced budget assigned to these schools. That is why it is PIE that produces, selects and evaluates the software in the first place, submitting the choice of software to several criteria, which are basically as follows. First, the program has to cover some of the curriculum content in the primary school. The second criterion is directed to the possibility of using the program for more than just one subject in the curriculum (that is, it has to be good for both mathematics and geography or history, for instance). Then, from those two criteria, a new evaluation is made in terms of a different set of characteristics, namely: (a) few material demands and (b) programs which are easy to deal with. As a consequence of these criteria, many programs which could very well make their way into school are ignored. As a matter of fact, Clic, developed by PIE, is one of the programs that fulfils all the established criteria, so it is, by far, the most frequently used program in the school and the one most used in the classes we watched. All this shows quite clearly that the school depends on the administration to develop any activity that implies using ICT tools, and this goes for the process of acquiring both the hardware and the software. The small budget causes the school to count on neither the best nor the most updated ICT resources and so the schools cannot do a great deal, especially when they deal with children from less than affluent families. In order to complete our approach to the first issue addressed in the research, we devote the next section to analysing what the software elaborated by PIE fosters in terms of learning.
ICT and mathematics 271 Analysis of the application: learning tasks and teaching. Clic (Busquest, 1992) is a software package that allows for the creation of new projects with content which can embrace virtually all the curriculum areas from some basic activities. A single activity grid supplies an output which serves a great variety of school subjects, thus making it possible to work at a very low cost. Also, the creation of new activities in this program is extremely simple, and the teachers have felt encouraged to develop them themselves, thus allowing for the tool’s even greater pay-off. The applications of this program developed for the teaching of mathematics were the most frequently used during the school year. They feature the presentation of information which has then to be applied in the resolution of exercises. In fact, the program can be classified as the typical ‘drill and practice’ kind of exercises, where the students either have to complete the information given, or select it from among the multiple choice presented to them. Our analysis followed the teaching and learning categories described by Kemmis et al. (1977). These categories are summed up in Tables 2 and 3. The group of activities surrounding the use of Clic which we had a chance to observe and analyse were examples of the instructional model. The cognitive interactions activated were Recognition and Recall types. Table 2. Learning tasks (Kemmis et al., 1977) Cognitive tasks in learning Recognition Recall Reconstructive understanding or comprehension Global reconstruction or intuitive understanding Constructive understanding
Information that has been presented before and that has to be recognized. The students correctly remember the information previously presented to them. The students have to reproduce the information taking into account its meaning. The students have to use the information that has been presented to them in the resolution of a problem, for which they need to reconstruct its semantic structure. The students are able to propose new problems from the information that they have over a given area of knowledge.
Table 3. Teaching models (Kemmis et al., 1977) Teaching models Instructional Revelatory
Conjectural
Divides the learning activities into smaller units. It focuses on positive (or negative) feedback of correct (or incorrect) answers associated to these activities. The structure of a subject reveals itself to the students, whether from logical organization, that the students discover either by interacting with it, or from developing different activities that are proposed to them with that purpose. Knowledge is built through the manipulation of ideas and the contrast of hypotheses.
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The children were presented with some material and then a series of exercises was proposed which were somehow meant to activate the information that was previously presented to them with a gradual increase of difficulty. The program had no means of finding out the way in which the children solved the exercise because it only verified the answer, which was correct if it coincided with the one determined by the program. It is possible that some children activated some knowledge, or rather the understanding of the concepts implied to resolve the activities, but it was not the program that stimulated this type of cognitive task. In the next section we shall consider the second dimension of the problem, looking at the mediation performed by the program when it was used in class, how it interacted with the teacher’s ways of teaching, and what influence it had over the exchanges produced in class and in the children’s learning. The computer in mathematics curriculum development The mathematics curriculum, in both its written established form and that which comes alive through the development of classes, stresses ‘procedures’, that is, the acquisition of: a set of skills, strategies, rules or guidelines for action, routines and ways of doing things, tactics and methods, algorithms, etc., . . . through which (the student) will become competent, practical, maybe an expert—depending on the type and degree of the proposed learning—in significantly facing his or her environment. (Valls, 1989, p. 34)
Nevertheless, this attitude gradually became limited, through the unfolding of the analysed maths classes, to doing ‘exercises’ or ‘problems’ completely unconnected to the students’ daily environment. The procedural approach was replaced by the mechanical performance of different kinds of exercises, where obtaining the correct result was more important than understanding what kind of everyday problems it could respond to. The program most often used in these classes, Clic, was in agreement with the teaching proposal of the mathematics teacher. In fact, when she considered the selection of logical support programs, she took into account, in the first place, that they dealt with topics contained in the mathematics curriculum and, in the second place, that the ICT approach would not be too different from what she was planning to do in class. Computer-assisted maths classes: a basic pattern. These classes were different from regular classes in that they were organized around interactions of students with the computer, not with the teacher. The teacher would only step in when a student required her intervention or when she considered that her participation could contribute to the resolution of the exercises. This characteristic could be seen from the basic outline of the classes: 1. Spontaneous and/or teacher-oriented organization of the work teams. 2. The students occupy their places and turn on the computer.
ICT and mathematics 273 3. The teacher gives information about the program and the set of activities to be carried out. 4. The children start working, the teacher walks among the groups supervising their work, the students ask for help when they don’t understand a proposed activity. The change of interaction pattern observed in computer-assisted classes coincides with findings pointed up by several researchers (Swan et al., 1991; Goodson & Mangan, 1995). Most studies show that in such classes the time spent on teachers’ explanations, traditionally given when introducing new subject matter, as well as their questions, are reduced to a half, at least. On the other hand, the activities in small groups of students tend to double in comparison with regular classes, where the teachers become facilitators (Chattrabhuti, 1986; Fraser et al., 1988; Hoyles & Sutherland, 1989; Stevenson, 1989; Somekh, 1991).2 According to Goodson & Mangan (1995) this pattern can extend to different subjects, including the ones which follow a very teacher-centred pattern in the classroom, such as history or geography, for instance. Nevertheless, the fact that the teacher participates less does not necessarily mean a learning improvement or a higher degree of student autonomy. It could happen simply because it is the computer that is now taking the teacher’s leading role in providing information and asking questions. Student–teacher interaction. Both the concepts of ‘the zone of proximal development’ (Vygotsky, 1979) and of ‘scaffolding’ (Bruner, 1988) were instrumental in our analysis of the teacher’s contributions. She constantly tried to provide elements which would help the children solve the problems, either through questions or through the presentation of helpful information. In most cases, this help was decisive in enabling the student to solve the problem. The teacher interacted with each individual student in relation to the difficulty that the child raised at a given moment. Her strategies were quite varied. Sometimes they consisted of reminding the child of the meaning of a concept, which would enable the student to know what he or she had do in order to solve the exercise that was currently being worked on. At other times she reminded them of problems of the same type, involving similar procedures. In one particular class, for example, algorithms were applied, and it seemed most important to retrieve the knowledge or the strategies which would enable the students to apply them successfully. The fact that students solve a particular exercise does not, however, mean that they have developed a genuine understanding. In fact, this did not seem to be the aim of the class. The categories of Edwards & Mercer (1988), presented in Table 4, show the analysis of teachers’ interactions. They show how the teacher may promote a ritualized knowledge, that is to say, the knowledge that substitutes for the understanding of underlying principles. Knowledge eliciting. The questions and direct enquiry on the part of the teacher were intended to verify the comprehension on the part of the student, but above all
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Table 4. Categories of the teacher’s interaction in class (Edwards & Mercer, 1988) Eliciting knowledge
Signalling significant knowledge Reformulating
The teacher’s questions or direct enquiries are meant to confirm the students’ understanding but, above all, they are directed to avoid transmitting information, getting it from the students instead. This kind of interaction usually implies the emphasis on key words, or the repetition of statements. The teacher repeats the student’s answer but he or she reformulates it so that it results in the correct idea in terms of his or her own definition of the knowledge.
their purpose was to avoid transmitting information, and instead ask the children for it. This is the most common type of question that occurred in class. For example, in questions like these: Teacher: If you multiply by 10, how much is it? Is it something like 297?. . . (the child gives an affirmative answer) . . . so, you could multiply by . . . (the exercise is 27 6 ? = 297).
The student was induced to give an answer he or she hadn’t thought of, that is, the teacher obtained an answer from a question that required the student to solve the problem, to use a procedure. However, there was no indication in the teacher’s speech that there might be a concern to go beyond the resolution of the problem or to have the child explain why answering this particular question could be useful to deal with this problem. Signalling significant knowledge. This type of interaction usually takes place by either stressing some key words or by repeating some statements in which the knowledge that is considered valid is emphasized. In the case of the classes that we watched, this kind of interaction could be seen when the teacher repeated a question either because the student gave a wrong answer or failed to answer altogether: Teacher: If you have 39 sweets and you only have to give them to one person, how much are you going to give him or her? If you have 39 sweets and you only have to give them to me, how many are you going to give me?
The teacher asked the same question again. We do not know whether the student did not understand it or simply did not relate it to the problem he or she had to solve. Rewording the question in this case was almost equivalent to giving him the answer. Nevertheless, there was no sign that the child understood why he should divide by one. Reviewing. In these classes, reviewing took place when the teacher had to explain concepts which had already been worked upon in class. In the examples given here, the teacher is reviewing when she explains when a number can be divided by five: Teacher: When is a number divisible by 5? (child answers). Teacher: When is it divisible by two? (child answers).
ICT and mathematics 275 Teacher: And by 3? (student doesn’t know, the teacher helps). Teacher: If the sum of the numbers is 3, 6, 9 or 12, that is, if the sum is some multiple of 3, it is divisible by 3. If it can be divided by 2 and by 3, it is divisible by 6. . . (the rule has to be applied to number 492) (the student fails to resolve this, and so they go over the entire process). Teacher: You know that if it is by 3 and by 5, then it is by 15. In that case you have to choose 15, then, is it by 3? . . . (student answers). Yes, why? Because the sum is a multiple of 3 . . . is it by 5? Yes, because it ends in zero, so . . . is it by 15?
In the first example, the teacher provides information assuming that the child knows it. In the second example, she herself answers the questions that she poses. In this case she is not re-formulating the student’s speech; she is dealing with her own. These interactions are similar to those given in regular maths classes. This fact is not surprising. It is consistent with the sense that the computer has come to fulfil the same role previously occupied by the teacher and/or the manual, which were the main learning mediators in ordinary classes. The ICT tool is assimilated to the way in which the teacher focuses her teaching. ICT only stimulates a greater interaction among the children. Interaction among the students. The fact that the computer becomes the centre of the class provokes a greater degree of interaction among the children, as boys and girls help each other in the resolution of the exercises. The categories elaborated by Mercer et al. (1991), which we present in Table 5, enabled us to analyse the interaction among students. These categories are relevant, since they were built on the basis of the SLANT research project (Spoken Language and New Technology) about computer-assisted teaching. The main objectives of this project were: (1) to identify ways in which computer-assisted teaching can provide a context for exploratory and argumentative conversation; (2) to describe the amount and quality of the conversations that result from computer-mediated activities; and (3) to provide information about the teacher’s role as supporter and mediator of these activities. The research was based on different studies of computer-assisted learning. It not only enlightened us about the types of interaction that took place in class, but also established relations within the interaction, the teacher’s role as an activity organizer and the role of educational programs in promoting certain kinds of exchange. Table 5. Categories of interaction among children (Mercer et al., 1991) Disputational talk
Cumulative talk Exploratory talk
It is characterized by disagreement between the participants and individual decision-taking: brief exchanges that consist in statements and arguments around doubtful points or negatives. The speakers build in a cumulative way, that is, they base their contribution on what their partner has said before, without being critical. The speakers deal with what their partners have said in a critical but constructive way. Statements and suggestions have to be questioned, doubtful points have to be justified, and alternative hypotheses have to be forwarded.
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Disputational, cumulative and exploratory talk. In order to carry out our analysis, we borrowed Mercer et al.’s (1991) three categories of interaction. The interactions which took place in the classes that we watched corresponded mainly to the first two categories. Most of the exchanges occurred when a child helped another resolve an exercise. This kind of help was limited to providing a cue, and sometimes a direct answer resembling the teacher’s. These exchanges can be considered disputational. Sometimes the conversation would become cumulative: one of the children would offer his or her point of view regarding the resolution of the exercise and the other child would build upon his or her partner’s contributions without discussing them. An exploratory discussion is one type of interaction related to the discussion of ideas, where each participant has to justify his or her own ideas, as well as criticize or find the weak points in his or her partner’s. This kind of discussion did not take place in the classes that we watched. According to the perspective used to appraise learning in this study, this constitutes a hindrance, since it is particularly in social situations of exchange, such as daily real-life situations, where we are most often engaged in mutual exchange with others, which favour learning the most. The knowledge of the group is greater than the individual’s and plays an important role in thought configuration. In this sense, the fact that exploratory discussion did not take place among the children when they interacted with the computer is an aspect on which we should reflect. We discuss this a little further in the next section, where we consider the influence of different kinds of logical support on the kind of interaction it produced. Types of interactions and software. The type of logical support that was used proved to have a great influence in promoting a certain type of exchange among the students in project SLANT (Mercer et al., 1991; Mercer, 1994, 1997). In the situations that were studied, closed programs tended to promote a type of interaction that did not favour the discussion of ideas. In fact, the exchanges produced among the children in the ‘drill and practice’ classes never went beyond cumulative chats. By contrast, the occasional use of an open program, the microworld Winlogo, confirmed the connection between the choice of program and the students’ interaction pattern. When the children were faced with unexpected results of their actions, they started to advance hypotheses on why the turtle-shaped cursor had moved this way or the other, in order to correct its position through some new command. Before responding to the activity proposed by the program, there was some discussion in order to choose the most appropriate course of action. Thus it was the group that was in charge of giving the start; there was an answer, usually an unexpected one; a new hypothesis was built and a new interaction took place. In these exchanges, the children used their previous knowledge, tested it, and then they built hypotheses that progressively brought them closer to the resolution of the proposed problem and the elaboration of new ideas. This was one type of interaction that agreed much better with the promotion of learning through construction than with passive reception.
ICT and mathematics 277 Nevertheless, it was Clic that was mostly used in the classes that we observed and basically the interactions among children were of the same kind as the ones that took place in ordinary classes, although their number increased in the computer-assisted classes. We will now present some notes on the learning strategies that the students used in order to solve the activities surrounding the Clic program, and the students’ interaction with the computer. Interaction with the computer: learning strategies Regarding the students’ interaction with the Clic applications used, we observed several strategies for the resolution of activities. They depended on individual differences among the children, and they were more or less oriented either towards mastery or towards action. As is often the case with classifications, when confronted by the complexities of life, some activities comprise a combination of strategies and so boundaries are often blurred (Prawat, 1996). All in all, the learning strategies used by the students while we observed them roughly corresponded to the categories of: trial and error, successive approaches, and reflection on the activity. Learning strategies: trial and error, successive approaches and reflection on the activity. Almost all students used trial and error for the resolution of the activities presented by the Clic program. From a motivational point of view, they were children geared towards action. They aimed to solve tasks mostly in order to obtain a positive appraisal on their competence. Children who belong to this group seldom wonder what to do before solving a task: they attribute their success or failure to chance and they see their own errors or the remarks of the teacher rather as punishment than as guidelines to modify their conduct. Such was the case of a boy with some comprehension difficulties, who enjoyed working with the computer and was much more motivated than in regular classes. He would refuse to make any effort to think about what he had to do in order to solve the exercises and the program did not offer any kind of resistance to this kind of conduct. Nevertheless, not only the action-oriented children tended to show this type of behaviour. One girl whose attitude in the regular classes enabled us to define her as mastery oriented also turned to trial and error in the Clic activity packs, particularly when the task became difficult, when she did not know how to carry it out or when she had not read the definitions that sometimes preceded some group of activities. Sometimes the students tried successive approaches. This strategy is situated between reflection (thinking effectively what they have to do, building hypotheses and confirming them) and trial and error. The children solved the exercises that they thought were the easiest first, the activities they already ‘knew’ how to solve, reducing the amount of possible options, in order to give an answer to those they considered the most difficult or which they did not know how to solve through trial and error. Finally, some children, no matter what type of exercises they were presented with, were most often inclined towards mastery. They did not want to solve the exercises by
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making use of trial and error strategies, or successive approaches. They wanted to make sure of what they had to do and why they should solve problems one way rather than another. Conclusions and perspectives In the case that we studied we could see that incorporating the computer into the school represented a great effort. If we consider the school’s circumstances, one trait of great relevance is that installation of the computer depended on the administration, as regards the supply of both hardware and software. Tight budgets and the need to reduce costs were responsible for the insufficiency of equipment and also for the reduced selection of educational programs. Obviously a solution would lie in increasing resources. Nevertheless, a mere increase in ICT resources in itself would not advance learning and teaching processes. On the other hand, the fact that the computers were kept separated from regular classes, with few access opportunities, also affected the benefits that could be obtained from their use. A less rigid management of space and time would allow for a more flexible use of the computers. In the case we studied, two aspects seemed to be modified in the mathematics classes that differentiated the regular classes from the computer-assisted ones. On the one hand, the class organization was less centred on the teacher: the students often decided when the teacher had to step in. On the other hand, at least 80 per cent of the children worked with a classmate, which was something rarely found in the regular classroom. In spite of these changes, the analysis showed that other aspects, like teacher–student interaction, suffered little change. The cognitive tasks that prevailed were the ones promoted by the software that was being used—now the centre of the class—which in general corresponded to the lowest cognitive requirements and memory demands, even if the children, guided by their personal characteristics, performed comprehension activities as well. As a matter of fact, the improvement that we could detect corresponded to McCormick & Scrimshaw’s (2001) most rudimentary level of change, consisting in performing more efficiently what was being done before. On the other hand, while the students who were rather mastery oriented tried hard at comprehension tasks, the action-oriented students, although they were much more motivated, continued to be just as ‘lost’ in terms of learning. In general, the teacher did not induce discussion or debate among the children to solve the activities, and neither did the program. Therefore, serious group work geared towards complex types of interactions fostering reconstructive, not to mention constructive, types of understanding, was greatly neglected. All this confirms the views of Cohen (1988) and Cuban (2001) in the sense that the implementation of a new tool in the school does not necessarily imply any innovation. How could education in general, and the teaching of mathematics through computers be improved? According to McClintock (2000), who tried to find out about the innovating potential of ICT in schools, it is factors more related to the
ICT and mathematics 279 structure of the school and the ways it carries out the teaching and learning process which would allow us to take advantage of the potential of these tools for improvement. Actually, the school itself should change if we are to profit from such versatile tools as computers. This leads us to believe that the route to future investigations should take into account the study of the purposes of current educational policies regarding the introduction of ICT in schools. Nevertheless, it is crucial that any study or measures that consider the implementation of ICT should take strong support from the vast supply of elaborated knowledge already in existence concerning widespread innovation in school and how it can come about. Notes 1. These works are quoted by Gallego (1992). 2. These works are quoted in Scrimshaw (1993) and Wegerif & Scrimshaw (1997).
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