Arithmetic Problem solving with ARI-LAB: an hypermedia multiple-tools + com.syst
We refer to the use of the ARI-LAB system with two classes of primary school (7 years old children). The ARi-LAB system, which was designed to develop arithmetic problem solving capabilities, combines the hypermedia technology with the technology of communication systems, both in local network and in remote. The system consists of different connected environments that allow the user different kinds of activities: to interact with a number of visual representation microworlds, to access data-bases of solved problems, to cooperate with other students or with the teacher by exchanging solutions (or part of solutions) and verbal messages.
The mediation of ARI-LAB allowed the subjects to tackle and solve additive and multiplicative problems harder than those they were able to solve before the beginning of the experiment. Pupils became acquainted with the system after few sessions, whereas their autonomy in the resolution process rapidly increased during the experiment and became satisfactory in the last sessions.
Education, mathematics, arithmetics, problem solving, educational computing, hypermedia, communication systems, cooperative work, learning tools, teaching tools, semiotic systems, microworld, visual representation systems
Discipline, subject : mathématiques
Public : primaire
Sek I, BWK
lower high school
upper high school
Bottino, Rosa Maria
C.N.R., Istituto per la Matematica Applicata, Via De Marini 6
Fax : 39 10 6475660
Pédagogie, pedagogy : The project consists in the experimentation of an innovative method aimed at introducing and developing arithmetic problem solving abilities at primary and secondary school level. The project is based on the use of the ARI-LAB system. In the last three years several classrooms were involved in the project. For the purpose of this report we consider in particular the work carried out by two ordinary classes of primary school (age 7). The students were proposed a number of arithmetic problems which they have to solve through the interaction with the different environment of the ARI-LAB system and the communication with other students or with the teacher, communication which is also supported by the system. From a mathematical viewpoint, the problems proposed to children are additive and multiplicative problems requiring different strategies (e.g. completion, total/part/remaider, containment, partition, ...). These problems were concerned mainly with buying and selling. Contrary to what would happen in paper-and-pencil problem solving, at the beginning the students almost never wrote down arithmetic symbols in order to express quantitative relationships among the elements of the problem, but rather they used the visual representation systems of the microworlds of ARI-LAB (for example, they could generate coins, move them on the screen in order to group them in meaningful ways, and so on) in combination with written verbal language. Only when a resolution strategy had been found students are, in some cases, requested to describe their strategy by means of arithmetic expressions.
From a pedagogical point of view the used system has the following characteristics:
= it provides a set of microworlds in order to represent a wide range of problem situations and, at the same time, to focus different mathematical structures and perspectives;
= it may deal with both real life and standard arithmetic problems;
= it is oriented to the design of resolution strategies within the microworlds and not to the development of computational skills;
= it is aimed at promoting the construction of content-dependent strategies and fostering the development of arithmetic concepts and processes based on the mastery of the meanings involved in the problem situation;
= it allows pupils to build resolution procedures starting from strategies based on the manipulation of symbolic objects visualised on the screen;
= it provides a set of solved problems in order to promote analogy-driven strategies and prevent pupils from getting stuck;
= it allows actual interaction between groups of pupils; this means that pupils can communicate to each other their tentative resolution procedures, including parts containing non verbal information, and to keep track of the messages.
Apprentissage, learning : The work in the classrooms is based on our idea of what it is to be considered as 'arithmetic knowledge' at the age at issue. Our assumptions take into consideration both our experience and the results of mathematics education research about how to develop arithmetic learning.
Arithmetic knowledge includes a lot of procedural skills, but cannot be identified with the ability to perform written calculations. It should be related to the ability to use arithmetic concept, procedures and symbols in order to solve problems. The problem solving process requires a suitable representation of the problem situation and the design of a resolution strategy. The ability to solve arithmetical problems may be viewed as the ability to realise expressions incorporating a meaning suitable to interpret the problem situation. This ability depends on the wealth of meanings that the subject can associate with the symbols of the representation system used. Arithmetic symbols are abstract and require to be associated to meanings expressed in semiotic systems which students are able to control. If pupils have not yet developed these abilities, they might use symbols without being able to give them any meaning and thus a sort of contract is implicitly stipulated between the pupils and the teacher, according to which to solve an arithmetic problem means to guess the correct operation and to perform the computations involved.
Therefore the task of the work in the classrooms was to allow pupils to deal with a wide range of problem situations and to construct specific strategies to obtain a correct answer, not only starting from their mathematical knowledge but also from their knowledge on the problem situation.
Enseignement, teaching : The teacher intervened during the problem solving activities both directly or through the communication via computer. In both cases the ARI-LAB system offered the teacher a means which effectively supported the communication with the student giving her/him means (e.g. visual representation and actions opportunities) through which she/he can mantain a meaningful "contact" with the students and, at the same time, can offer suggestions to the user which are of the same nature of the solution procedure she/he is performing.
From the point of view of the teachers it was a very interesting and useful experience since it gave them the possibility to make experience with a way of teaching arithmetic which is different from the one they usually perform. The main goal of this kind of work was the construction of meaning for arithmetic symbols rather than the development of computational abilities. Moreover, the experience gave the teachers also the opportunity to see technology, and computers in particular, as teaching supports which are not aimed at substituting them, but at helping them both in the communication with the students and in the tools they make available.
Technique : The ARI-LAB system is implemented in HyperCard 2.1 using HyperTalk programming language. It runs on Apple Macintosh computers with at least 12 inch monitors. The school were the project was carried out was equipped with 8 Mac LCII connected through a local network.
ARI-LAB is a hypermedia system which consists of a structured and connected set of environments which offer the users various options in order to solve a given arithmetic word problem. Two different kinds of users are expected: the student who has to solve a given problem and the teacher who can configure the system according to the needs of her/his students. The student interacts with four main environments: the visual representation environment, the strategy building environment, the communication environment and the data-base environment. The teacher, in addition to the environments available to the students, can access another environment, the teacher environment, which allows her/him to set different layouts of the system according to her/his educational goals.
The user interface of ARI-LAB, which is entirely mouse driven, is designed in order to make easy the passage from an environment to another, to automatically perform the dynamic increment of the system and to take account of the changes caused by user's actions. Starting from a main display frame, where the user can choose if she/he wants to solve a new problem or to see the solution of previously solved ones, different windows are presented according to the choice made.
Each window corresponds to a different environment. In the strategy building environment the user choses an arithmetic problem, from a set prearranged by the teacher, and builds her/his solution step by step using different kinds of languages (written, visual, symbolic). At each step of the resolution process she/he can access the visual representation environment which consists of a number of different visual representation microworlds which are provided in order to allow to the user to represent a wide range of problem situations and, at the same time, to point out different mathematical structures and perspectives. The representation microworlds available at present are: abacus', coins', spreadsheet', calendar', histogram maker', measurement division', partitive division', art bits'. Within each microworld graphic animations are available. Through interaction with the microworld chosen, pupils can activate different animations until they get a visual configuration representing a resolution step they recognise as adequate to their purposes in relation to the problem situation. Pupils can also partially or totally paste their productions into the strategy building environment..
At any moment of the interaction with the system the user can access the communication environment which allows her/him to insert the production of a solution into a social interaction process; in this environment different kinds of interactions are possible: the student can send or receive messages and solved problems (or partially solved problems) to another student who is in contact via a modem connection; or the student can be in contact with other students of her/his same class connected via a local network. The communication environment is also accessible from the data-base environment.
Users can access the data-base environment when they want to see problems previously developed and classified by the teacher or problems they had previously saved (their own problems or problems received from other users).
Société, society : Due to the characteristics of the system used, during the experiences the problem solving process had been inserted within a social practice that changed students' attitudes towards the problem, their assumptions on how to solve it and the validation situation in which the resolution process was inserted. The resolution process develops in a communication practice including the interactions not only between pupil and teacher but also among pupils; co-operation and negotiation processes are thus enhanced.
Different forms of cooperative work were tested: exchange of solutions between students; cooperative search for a solution by two or more students; exchange of comments, criticism or questions about the solutions proposed.
A way of using the communication environment was to propose problems to students which were different from those usually afforded in standard lessons. For example, we proposed to two students, who are in communication through the system, different but complementary problem texts in which each of the students has to play a different role (e.g. buyer and seller). The solution of the whole problem situation requires an integration of the information given in each text, integration which can be performed through an interaction activity exploiting the opportunities offered by the system. As tests pointed out, the presence of a real interlocutor allows a better understanding of the problem situation, since it fosters in the user anticipations about the role assumed by this interlocutor within the problem. Moreover, communication activities help the development of resolution strategies since each user can turn to her/his interlocutor as a resource which can give her/him hints, feedback and information in relation to the actions performed.
Another way of using the communication environment which was investigated during the experiences was to use it in the comparison of strategies between students. The mediation of ARI-LAB proved helpful in this context since it supports the comparison activity according to the following useful features: availability of different representation systems through which a solution strategy can be presented in a non-ambiguous way, sharing of solution resources, opportunity of integrating different communication forms, e.g. real time dialogue and exchange of solutions. According to these features, each user knew that her/his interlocutor could easily understand a produced solution and could also reconstruct the way in which it had been obtained.
Culture : The way in which the ARI-LAB system had been used in the work in classroom put in evedence the processes through which significative aspects of culture (knowledge, behaviours, tools, languages, rules meanings related to arithmetic problem solving), which at the beginning of the educational itinerary are under the control of the teacher, can pass under the control of the student.
Students had the opportunity to acquire a mastery in the use of semiotic systems through which they can model the solution of an arithmetic problem in a way close to that involved in real social activities.
Institution : During the experiences each student had at her/his disposal a computer (Macintosh LCII). To meet this requirement in one case it was necessary to divide the class into two groups. Each group used the computers at different times. Computers were connected to each other through a local network. The computers were placed in the computer laboratory. The work was performed during normal school hours.
The evaluation of the experiments is based on the analysis of the observation protocols taken by the teachers and by the researchers who constantly followed the experiences, the records of pupils' solutions and the dialogues held via the communication environment.
The experience in each classroom consisted of about 15 working sessions of about 1 hour and a half each.
Logistique : The teachers have to agree with the general philosophy underlaying the design of the system and its use in classroom. We have experienced that the proposed work had modified the characteristics of the didactic contract taking place between the teacher and the pupil in the solution of an arithmetic problems, for this reason it is necessary that the teachers are conscious of this fact and agree to modify their teaching practice consequently. We observe that no previous notions about computers are necessary both from the teachers and the students.
The familiarization with the system had taken a very short time (about one hour) and we have experienced no problem in the use of it throughout the work. A good mastery of the system was gained even by pupils who in general exhibit serious learning problems (e.g. a girl who was classified as having special educational needs)
Remarques, remarks : After the first experiment in classroom, we have added to the system another facility (the monitoring facility) in order to evaluate students' behaviour: the system now can automatically record anything a pupil has produced within a microworld in a working session, and then even anything she/he does not want to paste in the strategy building environment or she/he cuts before quitting. At the end of the session the teacher could review the sequence of the steps performed, in a sort of movie. This feature had proven very useful for the evaluation of the experience and to gain a deeper knowledge about the way in which students perform the different resolution strategies. Moreover it give the opportunity to point out when difficulties arise and where crucial point are.