We distinguish between internal representations and external representations, using the same terms used by B.Dufour-Janvier [8], but in a wider connotation.
In Exploiting Mental Imagery with Computers in Mathematics Education, R. Sutherland and J.Mason eds., Berlin: Springer Verlag, Series F, vol.138, 1995, pp.20-33. ISBN: 3540585826
External Representations in Arithmetic Problem Solving
Giuliana Dettori and Enrica Lemut Istituto per la Matematica Applicata, C.N.R., Via De Marini 6, 16149 Genova, Italy
Abstract: We discuss the role of external representations in arithmetic problem solving activities in elementary school (age 6-11). The analysis is made by referring to a curricular project which emphasizes the relationship between achieving arithmetic competencies and solving problems. First we analyse the role and the components of external representation in a pen-and-paper environment, then we discuss different characteristics and impact of using representations for arithmetic problem solving in a hypermedia environment. Keywords: External Representation, Arithmetic Problem Solving, Arithmetic Competencies, Elementary School, Hypermedia Environment
1
Introduction
Solving mathematical problems, as well as understanding mathematical concepts, very often involves building and handling representations. Representations are a powerful means for a person for communicating between his mind and the environment. They can be a tool to dialogue both with oneself and with others. We distinguish between internal representations and external representations, using the same terms used by B.Dufour-Janvier [8], but in a wider connotation. We call internal representation the whole set of mental images, thoughts and verbalizations that allow one to interrelate data, to distinguish main elements from secondary ones, to connect with one another knowledge on different topics and past experiences, to find out different possibilities and alternatives to be analysed, to concatenate reasoning steps. As concerns external representations, we extend the definition by Dufour-Janvier - "all external symbolic organizations (symbols, schema, diagrams, etc.) that have as their objective to represent externally a certain mathematical "reality"..." - including in it any sign a person can use to express a concept, a situation, a step in a thinking process, either written or verbal sentences of a natural or artificial language, and symbols, drawings or images, both with a widely accepted semantics or with a personal one. Internal and external representations have a fuzzy connection with each other, since the expression power of external representations of a person depends heavily both on his expression capabilities and on the effectiveness of his internal representations. At the same time, it should be recognized that there is usually some exchange relationship between the two levels, since the production of an external representation is usually 20
based on some idea, and, conversely, the contact with someone else's external representation gives rise to an internal one. In this paper we concentrate only on external representations, considering them not as mental processes, but as products of mental processes, and discuss the role of external representations in arithmetic problem solving, particularly aimed at the acquisition of the arithmetic concepts of number and elementary operations, which constitute the first and basic mathematical knowledge. First, we analyse forms and impact of external representations produced in a pen-and-paper environment (e.g. sketches), then we discuss different characteristics and impact of using representations for arithmetic problem solving in a computer hypermedia environment. We think that studying the influences of external representations on arithmetic learning can have a twofold impact: it can help the children in their learning process and can help the teacher to investigate learning mechanisms, working on concrete objects, yet remaining aware that interpreting a representation is a complex task. Solving a school problem always requires some form of external representations, at least to communicate the results. Also, it is usually necessary to use some more or less formal representation to describe both the data of the problem and at least part of the resolution strategy. Hence, it is necessary that the school encourage the development of at least basic representation abilities, taking into account that they are not spontaneous but have to be taught and learned. Learning to make useful representations from the first school years in a non-pictorial discipline like arithmetic is very important, since the ability to associate a mental image to abstract thought will turn out useful to students later, when facing higher levels of abstraction in other mathematical fields [5, 6, 7]. We developed some experience on this topic in a curricular project for the elementary school (age 6 to 11), where the development of representation abilities is constructed while pupils perform a problem solving activity in particular "experience fields" [1]. We found out that stimulating pupils to reason on external representations requires basic long terms educational choices concerning: the didactic contract; the instructions given to children, especially for the formulation of problem texts and the selection of numerical values; the choice of representation systems and the way they are introduced; the proposed behaviour models.
2
The Relationship Between Arithmetic Problem Solving in our Projects
Competencies
and
Learning arithmetic means learning to solve problems by building, applying and understanding strategies, and gradually starting to use suitable formal languages correctly and consciously. In our opinion, to reach these aims, it is necessary to establish a dialectic relationship between formal language acquisition and problem resolution, since the meaning of the language elements (numbers and operations) can be built only through the resolution of "real" problems, close to the children's life, hence meaningful to them. This means to give wide room to children constructing and interpreting resolution strategies rather than to learning rules. "Real" problems that we consider suitable to be proposed must belong to fields with strong semantic values [1], that is: they are meaningful within the chosen experience fields, they are as rich as possible from the point of view of involved mathematical structures; they can be gradually recognized by pupils as related to their extrascholastic experience; they allow structuring a not fake relationship between the child and the problematic situation and between the child and the requested answer.
These real problems on the chosen experience fields (the calendar, small businesses, etc), seem to be a necessary condition to induce into pupils representation forms which are connected with performed actions and produced reasoning. By solving these problems in different experience fields, children can gradually construct the different meanings of number (value, measure, ordinal, cardinal) and the different meanings of basic operations, together with some of their properties, which can be emphasized also by means of a voluntarily late introduction of written computation techniques. In the process of problem resolution, a crucial role is played by different tools for external representation, which allow one to represent, at least partially, concepts and resolution strategies. Hence, knowing and using suitable symbolic representation systems is one of the mathematical competencies which need to be developed. The use of these symbolic systems must be joined with a higher level of the mathematical knowledge which can be developed in these contexts and concerns a reflexion on the systems themselves (duality "tool-objects") [9].
3
External Representations in Problem Solving Activities
Making use of external representations in arithmetic problem solving is not automatic, neither in the choice of representation forms, nor in their use. Representations produced by children during a resolution strategy are based on the expressive codes they have at their disposal. These representations are not completely arbitrary but keep some analogical connections with objects which are involved in the posed problems. Since the representations for a given problem resolution are connected to its meaning, we believe that the preliminary step is to identify the three components of the problem resolution process in which external representations may be involved: 1) understanding the problem situation; 2) sketching the resolution strategy; 3) describing the computation strategy [10,13]. Concerning these three components, some observations are in order. Facing components 1 and 2, the child immerses himself into the problem, and the representations he produces can be so strong to act for him as new problem; in this case, the child solves his problem, rather than the given one, obviously getting wrong results if the two problems are not the same (in this case, representations become autonomous models [11]). Component 3 is dependent on the teacher's didactic choices concerning the search and application of computation strategies, in order to introduce written computation techniques only when the conceptual meaning of the arithmetic operations has been deeply acquired. Finally, regarding generality, representations developed for 2 and 3 above are dependent not only on the problem structure, but also on the data types and on numerical values required by the problem [2]. It will not be the case, of course, that, in every problem resolution, every child will draw external representations of the three components or clearly distinguish them from each other; in some case, some representation may remain internal (as can be found out by means of interviews). Moreover, the three components must not be considered chronologically ordered, but they interact with the resolution process in a way which is dependent on the child's cognitive style and on the complexity of the problem. However, distinguishing the three components constitutes a tool to analyse and reflect on the didactical activity and on children's performances, hence helping the teacher to activate a more attentive and accurate way to look at her pupils and to interpret their behaviour.
3.1 Interpreting the Problem Situation The meaning of the expression "representing the problem situation" may be obvious in mathematics fields like geometry, where problems usually have a pictorial aspect, but it is not as obvious in arithmetic. In the project we have been engaged in, which gives great importance to exploring and communicating one's thought, representing the problem situation means giving an alternative description of it, containing the information that the child considers important to solve the problem. Usually children draw a figure showing their interpretation of the problem situation, but this representation does not need to be pictorial. Being able to make an effective external representation for this resolution component means understanding the problem and becoming aware of it. It is a tool to proceed, not only to describe. After gaining resolution experience on some kind of problems, however, the pupils learn to work on the same problem situation without the help of external representations, and can proceed to the next components based only on their internal representations. 3.1.1 Lack in externally representing the problem situation Observing the behaviour of children, we noticed that the lack of external representations was probably dependent on the lack of internal representations resulting from two different states which are described below. The first state can be defined as blockage. The child does not know what to do, does not start any form of iconic, or verbal, or gestual representation, does not recognise any similarity with previously solved problems, feels anxious for having to produce a result. The problem data are blocked within the numerical data and have no real meaning, so that often the child, feeling pressured to do something, applies any operation (mostly the one he is able to compute better). Hence, he neglects the meaning of his actions on problem data and does not connect it with the considered problem. A second state is the automatic use of previously experimented resolution schemes [14,15] which are recalled by assonances in the text or in the structure of the problem. Without a deep understanding of the problem structure and of real analogies among data, assonances can lead to big mistakes (see example 8, §4.4). However, we want to repeat that lack of external representation of a problem situation does not always mean misunderstanding or lack of understanding. When a problem structure has been considered so many times to be absorbed with all its possible variants, representing the problem situation becomes one with representing the resolution and computation strategy. 3.1.2 Representing the problem situation In our experience, representing the problem situation is not easy for the child, nor does it imply that he will reach the correct solution. It can be ineffective (see fig.1), or even misleading [8] when it does not interpret the situation structure in a sufficiently complete and pertinent way, that is, involving data, relations among them and expected results, at least in implicit form. The difficulty of representing at this stage arises from the required mental operation of selecting among useful and useless data, implicit and explicit ones [8]. This difficulty is evident when the child represents the problem scene with great richness of useless features. However, though not independent of the resolution strategy, the representation can be important all the same, because it corresponds to the child's need to get hold of situations where the mathematical root of some operations can be found (e.g. buying and selling). It is part of a dynamic understanding of the problem situation and poses the
problem of how much it is necessary for some children to immerse themselves into the context before passing to more selective representations. This diversity of children behaviour reminds of how hard it can be to analyse their resolution tracks, understanding when their difficulties originate from bad communication or from incorrect internal thought. In our experience, children who are good problem solvers make representations which already contain germs of their resolution strategy, yet showing different personal styles. In this case, the representation is the result of an active intervention by the child to select meaningful variables and to put into relation the given data with the goal to be reached (Fig. 2).
Fig. 1
1
(1) (2) 20 = ......"
Fig. 21
"I have to put together these two numbers" (translation) "To know how much the used soil costs, I have to make this calculation 5700 :
3.2 Representing the Resolution Strategy In general, at elementary school level, it can be difficult to recognize the border between representation of the resolution strategy and representation of the computation strategy, in particular when working with scarcely explored operation meanings or with complex numerical data. This separation becomes more evident when solving arithmetic problems without numbers, or in a planning phase, before numerical values are assigned, since those kinds of problems stimulates general forms of reasoning and mental representations which are not limited by the reference to concrete numerical values. Hence, it is possible to make explicit the strategies connected with the memory of meanings and with planning abilities, as well as the "macrostructure" of the problem. Solving this kind of problems requires a wide use of verbal representations. In our experience, this kind of representation not only allows one to declare his reasoning, but supports production of hypotheses, simulation of trials, control of the meaning of the problematic situation and reasoning in its development (for instance, the terms "add" and "take away" are more suggestive for a 7-8 years old than the symbols "+" and "-" [2]). In problems with numerical data, it is more difficult to show the border between representation of the resolution strategy and representation of the computation strategy, but it is still possible - and cognitively important - to force a distinction between them. On the distinction between "I try to reach the number-solution" and "I decide what I have to do" is based the possibility to compare resolution strategies and to reproduce forms of reasoning in different contexts. 3.3 Representing Computation Strategies Representing computation strategies is more or less important according to the general teaching project. If the symbolic representation of operations and written computation techniques are introduced very early, this step receives scarce importance. It is the contrary in the project where some of us are involved, where those abilities are introduced only after exploring the various meanings of the operations themselves by solving many problems in "experience fields", since this exploration leads to better emphasize the meaning and some properties of elementary operations [1]. The representation systems for computation strategies are strictly dependent on the importance which is attributed to the various meanings of number (ordinal, cardinal, measure, value).
Fig. 3
Fig. 4
It is interesting to note that often the representation of computation strategies is accompanied by some pupil's written comments, which allow him to control his resolution process and to keep connected with the context (see fig. 3). Moreover, posing problems where the child's attention must be concentrated on the difficulties of the computation strategies constitutes a bridge towards the resolution of more complex problems with more than one operation. Indeed, switching back and forth when reasoning with numbers leads the child to check and to reshape his resolution process, and, at the same time, to consider the problem in its complex, mentally ordering the intermediate steps to reach the final solution.
4 Examples of the Role of Representations We analyse now some situations showing the impact of representations with respect to : 1) introducing or reinforcing operation meanings; 2) introducing new, higher level resolution strategies; 3) recognizing and correcting errors; 4) connecting the problem situation and its mathematic modeling. 4.1 Working on Operation Meanings Recalling measurement meaning of division. In this case, making a representation of the problem situation has been indispensable to starting a resolution strategy. Example 1 (4th grade, 9 y.o.). An elevator can raise up to 370 kilos. How many 12 kilos baskets can it raise at the same time? No child in the class, except Antonio, draws a representation, nor declares to have thought to one. Antonio, of low level, declares: "First I drew the baskets, then I understood that I had to divide to know how many times 12 kilos fits into 370". (After drawing the baskets Antonio erased them, probably in order not to look different from his classmates). Introducing the partition meaning of division. A representation proposed by the teacher has been able to unblock some pupils to solve a problem, stimulating a relationship among the problem data. Example 2 (3rd grade, 8 y.o.). A bottle of rennet costs 4,500 liras and contains 15 spoonfuls. We use it by the spoonful. How much costs every spoonful? The teacher proposes to blocked children a drawing showing a bottle with the indication of its price, and a spoon at its side. This representation could unblock some children, since it recalls the relation between volumes and costs and between bigger and smaller volumes. 4.2 Improving Resolution Strategies Constructing a particular representation of the problem situation, during its resolution, leads to modify the resolution strategy. Example 3 (4th and 5th grade, 9-10 y.o.). With 2,000 liras stamps it is possible to mail a letter that weighs no more than 250 grams. Maria has an envelope that weighs 14 grams, and several drawings sheets, that weigh 16 grams each. How many drawing sheets can she put in the envelope so that her letter does not weigh more than 250 grams? Some children verbally declare to use a trial-and-error strategy, and actually start working with it. After some trials they produce representations (fig.4) that suggest to subtract the weight of the envelope from the maximum allowed weight, hence obtaining a more efficient algorithm for resolving this problem.
A flexible, yet not arbitrary, use of representation systems that have been introduced in other contexts can activate new computation strategies. Example 4 (grade 1, 6-7 y.o). We want to compute how many holidays we had in April 1992, knowing that the following days were holidays: from 3 to 7: election days; from 16 to 21: Easter period; 25: national holiday; 12 and 26: Sundays. The calendar, used as a tool to order numbers and to count time periods expressed in terms of days, is used by some children as a counter: in the first column they write the numbers 1 to 30; in the second column they write the reason of the holiday, namely "election" in correspondence to numbers 1 to 5, "Easter" for numbers 5 to 11, "national" for number 12, "Sunday" for numbers 13 and 14. Hence the last number - 14 - is the answer to the problem.
4.3 Recognizing and Correcting Errors Suggesting the use of a more complete representation leads a child to revise the representation both of the problem situation and of the computation strategy. Example 5 (2nd grade, 7-8 y.o.). The teacher buys a bottle of dairy cream that costs 1300 liras. She pays with 2000 liras. How much change does she receive? In a first resolution attempt the child makes an incomplete drawing (a bottle of cream) and a meaningless computation (1300-2000=10300 in column alignment). The teacher helps him to simulate the situation and suggests a more complete drawing including the handed money (money drawing is a representation environment previously widely employed by the child). Working on the drawing rather than on symbolic representation, the child is able to solve correctly the problem. Requesting the child to reflect on the realized representation, leads him to decide which is an adequate and correct representation system. Example 6 (Nicola, 2nd grade but with bad learning problems, 7 y.o.). Today is December 12. How many days until the Epiphany , which is on January 6? Nicola tries a first solution: "I make the numbers: 13, 14, 15, 16, 17, 18, 19, 20, 1, 2, 3, 4, 5, 6 there are 14 days" (adequate but incorrect representation), then, after teacher's intervention, a second one: "13, 14, .........30, 31, 32, 33, 34, 35, 36 there are 24 days" (inadequate and incorrect representation), a third one: "13, 14, .......30; it needs to consider 30 days" (adequate but still incorrect representation), finally a fourth one: "13, 14, ..... 30, 31, 1, 2, 3, 4, 5, 6; there are 25 days" (adequate and correct representation ). 4.4 Connecting the Problem Situation and its Mathematic Modeling Not representing important elements of the problem situation can lead to macroscopically incorrect results even in presence of basically correct resolution strategies. Example 7 (5th grade, 10-11 y.o.). The class needs to buy some graph paper to finish some work. In the class there are 17 pupils, and each one needs 3 square pieces of graph paper measuring 1 dm2. The paper is sold in sheets, each measuring 21x18 cm. and costing 500 liras. How much is the total cost? Some children do not represent the paper sheets; many among them develop an incorrect resolution strategy, as follows: they compute the area of every sheet as 21x18=378 cm2 (i.e. 3,78 dm2), then assume that every sheet can contain 3 of the requested pieces of paper, without considering that 3 square pieces can not fit into one sheet. A too rapid activation of strong and efficient resolution and computation strategies can lead to misunderstand the problem situation and to give an incorrect result. Example 8 (3rd grade, 8-9 y.o.). In order to prepare egg noodles for twenty people, we need 15 hundred grams of flour and 15 eggs. We buy two 1 kilo bags of flour, but we are not going to use all of it. Compute how much we spend for the used ingredients, knowing that 1 kilo of flour costs 1,100 liras and that 1 egg costs 200 liras. Mariella will buy the ingredients for us. How much change will she get if she pays with a 10,000 liras note? All children correctly solve the first part of the problem, but, when computing the change, influenced by a previously applied scheme, they subtract from 10,000 the computed cost, without considering that we had to buy more flour than we actually used. The fact that they had already computed a total cost led them to forget the context.
5
Using a Hypermedia System to Develop Representation Abilities
At this point, it is natural to wonder what role the computer can play in developing effective representation capabilities and using them as cognitive help in arithmetic problem solving, since the computer is more and more used as a didactic tool and it has been actually found out that it can be a powerful means for manipulating verbal, symbolic and pictorial representations. Such a discussion is difficult to tackle, since the effectiveness of a computer system obviously depends not only on its potentialities, but also, and in particular, on its implementation characteristics, so that it is not possible to make an abstract analysis, without referring to some concrete system or environment. Hence, we base our considerations on a prototype problem solving environment, Ari-Lab [3,4], recently developed by some researchers of our Institute. The aim of this system is to help learning arithmetic problem solving by allowing the child to implement actions performed as anticipations of a future result, and to validate his actions through a dialogue with the machine. The system's operation is based on dialogues with the child carried out in three strictly interconnected environments: 1) visual representation; 2) strategy building; 3) communication (see fig.5). In the first environment, pupils work on the solution to a given problem by using, according to their wishes and needs, different "languages", choosing from a menu of hypermedial for the production of pictures. At present, 8 microworlds have been implemented: abacus, money, calendar, simplified spreadsheet, tables or histograms, division by partition, division by measurement, art bits. A representation constructed in this environment can be copied - in total or in part - in the strategy building environment, where it can be completed by verbal and/or symbolic descriptions. The communication environment, finally, allows communication and resolution's exchange between two children working on different computers or with the teacher.
Fig. 5 (from [4]) The Ari-Lab system is currently under experimentation, hence results on its use are not yet at disposal. Moreover, an exhaustive analysis of this aspect would obviously not be feasible in this paper. However, a first approach to its use led us to point out some basic differences between using a pen-and-paper environment or a computerized one. A first point is that the computerized system allows less freedom but gives the greater expressive power of dynamic representations. The limited number of microworlds at disposal to produce iconic representations is obviously due to technical limitations. This fact, on one side, limits both the types of problems that can be proposed and the kinds of representations that can be produced, which is negative for the children, but, on the other side, it can stimulate, at least in smarter ones, the cognitive activity of selecting more precisely the features to be represented and to find more abstract representation forms. Ari-Lab allows the use of more than one microworld in the resolution of a problem, but only one at a time, and does not provide any mechanism for working in a microworld on the representations produced with the previously employed ones. This limitation of possible microworld operations, on one side, can break the resolution flow of a problem, but on the other side, offers to the teacher an opportunity to help her pupils reflecting on the microworlds as objects. In any case, a system that communicates mainly by means of representations stimulates pupils to reflect, under teacher's guidance, on the importance of being able to interpret given representations and on the power of the produced ones. The fact that microworlds corresponding to experience fields - such as money or calendar - are proposed at the same level of arithmetical operations - such as the two meanings of division - though very reasonable in the economy of the system, may induce in the children wrong opinions about their relative importance. On the other hand, the contemporary proposal of different microworlds can both give a start to blocked children, and suggest to smarter ones to explore different forms of representation for a same problem. A useful feature of Ari-Lab is that the teacher can inhibit the use of some microworlds during an utilization session. In this way, the children are lead - or compelled - to use only one particular microworld, or to choose in a more limited number, which acts as suggestion for weaker pupils and as stimulus to develop higher level strategies for smarter ones. Other technological limitations are felt in the answering speed of the system - which is still a little too slow, at present, to keep children concentrated on a problem - and in the way new pictures are sometimes produced (for instance, in the microworld "hypermedia" new coins are produced in random positions) - which in some case can disturb the resolution plan under implementation by the child. These drawbacks, however, seem likely to be overcome in the near future. A greater ease in making computations can certainly lead children to disregard mental computation abilities, which, on the contrary, should be encouraged. Ari-Lab, however, partially compensates this problem by emphasizing the difference of different magnitude orders. In conclusion, using representations in pen-and-paper environment and in a computerized one seem to have different characteristics and different impacts.It appears that a computerized environment like Ari-Lab can be a useful support to learn to
construct and to use representations - also because creating and using automatically small drawings is usually appealing to children, hence motivating for them - but at the same time it can not be the only tool used to teach children arithmetic problem solving.
6
Conclusions
Learning to make external representations and reflecting on their use and on their characteristics is very important in arithmetic problem solving, both because they have essential unblocking and correcting effects on many children and because some forms of representation are integral part of the mathematical knowledge (like symbolic notation, histograms, graphs, tables, etc.). The various meanings of the concept of number and the variety of problem situations that can be faced require that the teacher accustom her pupils to different representation systems and to using them in the different components of problem resolution. Hence it is necessary to make pupils aware of the meaning, expressiveness, potentialities and limits of representations, and to guide them to choose every time the representation which is most functional for the resolution of the problem at hand. As concerns using computerized systems in this field, they can be useful for stimulating the children's representations abilities, but, with the present technology limitations, they can not be used as the only development tool. However, future developments of computer software may be more powerful in supporting representations, but the teacher's role will continue to be prominent.
Acknowledgements We are very thankful to G.Chiappini, R.M.Bottino and P.L.Ferrari for several discussions and for introducing us to their Ari-Lab system. We are indebted to E.Scali and A.Ferrara, teachers at an elementary school in Piossasco (Torino), who, for several years, carried on and synthesized an experimentation on representations in pen-and-paper environments. A helpful discussion with Richard Phillips is gratefully acknowledged.
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