1. Which problems?
1. "As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development. Just as every human undertaking pursues certain objects, so also mathematical research requires its problems. It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon." (Hilbert[1] 1902, p. 1).
2. "A problem arises when a living creature has a goal but does not know how this goal is to be reached." (Duncker[2] 1945, p. 1).
3. "Thus, a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking." (Polya[3] 1957, p. v).
4. "In my opinion, the problem is a group of words where there are numbers." (Lorenzo, a fourth year primary school pupil, in Zan 1991-92, p. 808).
By reading the four citations above, each of which either explicitly or implicitly refers to a definition of a problem, one may wonder whether they are really speaking about the same thing. They are four different points of view: that of a mathematician, of a psychologist, of a pedagogue, and that of a child (but, we believe, not his alone!).
Let us examine them one by one:
1. The point of view of the mathematician.
Hilbert's words characterize a problem implicitly, yet clearly. The focus is completely centred on mathematics. Obviously the person posing the problem is an individual (a mathematician), possibly the same person solving it or trying to do so. But once the problem has been posed, it becomes a problem for the whole community of mathematicians, that is it becomes a problem of mathematics.
2. The point of view of the psychologist.
Duncker's definition is totally similar to that of many other psychologists. It highlights that a problem is a product of the human mind. A task is not in itself a problem, it is if it is lived through as such. This definition emphasizes two fundamental conditions of a problem, both of which are centred on the problem solver. That there is a goal, i.e. the subject is motivated in reaching an objective, and that such an objective may not be reached through automatic proceeding.
The latter condition involves the classic difference between 'exercise' and 'problem'. The former, which obviously occurs when it is the very problem solver to pose himself the problem, opens up to a series of questions if the problem is posed by others (for example : 'Is the set objective shared by the subject?') and suggests the difference between 'task' and 'problem'.
3. The point of view of the pedagogue.
The two preceding definitions both appear to be rigid. The former is centred on the discipline, the latter on the subject. It is not easy to mediate between such two extremes, but this is exactly what needs to be done when the problem is inserted in a learning context. The problem referred to by Polya (who although a mathematician, presents himself as a teacher in 'How To Solve It') is the result of such a mediation. The focus of attention is both upon the discipline and on the learning subject. The psychological development and interest of the student are cared for, and the need to balance the difficulty to the acquired knowledge of the class students is underlined.
4. The point of view of the child (but not his alone!).
In the school context the problem is often something yet more different. It is a linguistic label which is 'attached' to certain types of exercises, generally those which present a well known, particular, fixed structure, and which are characterized by there being a text in which numbers appear together with a question whether implicit or explicit. Here, too, the focus of attention is on mathematics, not as 'discovery', but as a knowledge of rules to be 'applied' through repetitive exercises.
2. Mathematical Problem Solving
The term 'problem' persists in its ambiguity when we talk about problem solving activities. In fact research into problem solving contains a myriad of studies which vary enormously both in their object of study and in the methodology used. This multiplicity also depends upon the extremely complex nature 'per se' of problem solving, which involves a great variety of factors. An episode of problem solving may be considered as a small model of a learning process, and as such, as current research into the psychology of learning underlines, it involves not only cognitive but also metacognitive and emotional factors. The importance of such factors emerges above all when the failure of a subject, who appears to possess the necessary knowledge, needs to be interpreted. In particular, the beliefs of the subject may act as an obstruction and inhibit 'a priori' the decision or the capacity to use his cognitive resources.
Within the complexity which characterizes the activity of problem solving, the intervention of three deeply inter-relating independent variables may nevertheless be recognized (Kilpatrick, 1975, in Kulm[4], 1984):
If observed from a local point of view, each of these factors highlights a further complexity. Within such a complexity some of the significant features are suited for observation and control and therefore suggest a further emphasis on other variables. For example, if our attention is focused on the subject, the following may be recognized:
Task variables, instead, describe the more significant characteristics of the task, such as:
Finally the environmental conditions in which the subject solves the task are not intended to describe the task or the subject but are external to both. Thus they describe the physical, psychological and social setting in which the problem solving event takes place. The most important class within this category is represented by instructional factors (the problem solving instruction), i.e. by such conditions which may help a subject to improve his own abilities in problem solving (Lester[5], 1980).
The problem solving event ends however with a product which is influenced by the variables at play. Such a product is the explicit conclusion of a series of processes, activated by the subject, which can be either implicit or explicit, conscious or unconscious.
A further detailed analysis of the products and processes highlights significant variables. Product variables have to do with the achievement of the solution to a problem. This classification includes the time taken for a solution, the correctness or incorrectness of a solution, the completeness of a solution, but also the elegance of the solution or the multiplicity of different solutions found.
Process variables involve factors related to the individual's behaviour during problem solving, the euristic processes used, the algorithms employed, and the blind alleys encountered along the subject's path toward a solution (Kulm, 1984, op. cit.).
Naturally, the reality of problem solving, characterized more by the relationships between its own elements rather than by the elements themselves, is too complex to be constrained in whatever scheme. However, the proposed classification (adapted from Kilpatrick, 1975, and Lester, 1980, op. cit.) which we synthesize below, can be helpful to interpret most of the existing studies:
Even in simplest cases research into problem solving nevertheless involves the dynamics between at least two of the factors which have been highlighted. In general, attention is focused on one of the independent variables (the task, the subject or the environment), and how the processes or the products vary is studied, as one or more characteristics of such a variable vary. In this way one can study how the correctness of the result varies by modifying the characteristics of the task (e.g. by increasing the length of the text, or by writing numerical data in letters, etc.), or compare the products of subjects having different characteristics (age, sex, or even beliefs), or evaluate the effects of instructional treatments on the products. Also, one can inquire into how the resolutive processes put into action by a subject depend on certain features of the task, or on the characteristics of the subject himself, or on the type of education received.
Other studies however may concern the relationship among the independent variables shown in the triangle. An instance of this is the study of the influence that a particular type of problem, or the education received, have upon some characteristics of the subject, such as his beliefs or his attitudes.
We have already emphasized how limiting any scheme can be to describe such a highly complex activity as problem solving: even more if one keeps in mind that the researchers have the utmost autonomy in the choice of variables to be observed (or of the relationships among them), or, more generally, of the research methods to be used.
3. Research on problem solving and teaching: contributions in Italy (1988-1995)
Obviuosly not all the viewpoints chosen by the researcher in order to examine the complexity of the problem solving activity have the same effects on the school context.
This remark leads to a question of general type:
- Which research studies on problem solving (i.e. which points of view and which contexts to observe) can offer the best contributions to the teachers?
Even more importantly:
- Does theoretical research on problem solving have any effect on the school context? If so, which?
Questions of this type are of special interest in the Italian context, since this appears to be somewhat peculiar. The research into mathematics teaching in Italy has been organized into the 'Nuclei di Ricerca Didattica' (NRD), under the auspices of the University departments of mathematics. Yet, whilst those scientifically in charge of such NRD's are always University teachers, other members are teachers from different types of school, whose daily activity is school teaching in regular classes. Italian research is mainly based on their goodwill, and on their being both capable and professional. Such teachers, called research teachers, are not officially acknowledged by the Italian Administration, and do their research voluntarily, in addition to their school commitments. This entails that the application of this research to the practice of teaching, as pressingly asked by such teachers, cannot be eluded.
Beside the above remarks, it is worth emphasizing the special attention to problem solving paid in the official Italian national curriculum, mainly in the primary school. Here not only is a general educational peculiarity called for, but concrete details in teaching are considered. (A detailed presentation of problem solving in the national curriculum of the Italian primary school may be found in D'Amore, 1993a).
Going back to the two previous questions, in order to answer the latter, it may be observed that in school practice there is often a lack of effective problem solving activity. The definition given by the child which we quoted at the beginning ("...the problem is a group of words where there are numbers") reflects his beliefs on the nature of problems drawn from his school experience. There is a deep difference in meaning, therefore, between the problem seen as a routine excercise, though with certain formal features, and the problem as the object of most of the research into problem solving. This difference has several implications which make it difficult to use the results of the research on problem solving in teaching practice. In fact, the role of the problem in teaching, as well as the roles of the teacher and the subject, are different in the above two cases.
Whilst in some cases the problem-solving activity is still reduced to simply doing routine exercises, in other cases its value is recognized by teachers, who are generally aware of the potential that problem solving represents in learning mathematics, and who tend to become more sensitive to the fact that a problem solving ability is an objective which is relevant in itself. Particularly significant from this point of view are instructional studies, i.e. those research studies having the object of developing in students the ability in solving problems.
Hatfield (1978, in Lester, 1980, op.cit.) has distinguished among three types of problem-solving instruction:
The paradigm of traditional experimental research on instructional studies includes:
This type of research is characterized by the accurate control of significant variables and the consequent breaking up of the problem-solving activity into a great number of component factors. It shows however intrinsic limits in characterizing the successful teaching of problem solving in the classroom (see Silver[6], 1985; Boero and Ferrari, 1988). The role of factors which elude control and measurement appears ever more determining in the teaching-learning process. The need emerges to keep under consideration the profound interaction among the various elements involved in the learning process (see Lester[7], 1994): for example teacher-student interaction, or student-student interaction, but also the type of classroom atmosphere that exists. Under such a light, a research studying problem-solving in the classroom as a whole, rather than its single components, might be more opportune. Beside traditional studies of experimental research (which are characterized by an accurate control of variables at play), 'teaching experiment' studies introduced by Soviet research (see Kantowski, 1978, in Lester[8], 1985) are acquiring an increasing importance. These are characterized by the following features:
1. They are non-experimental in design.
2. They take place over an extended period of time.
3. They attempt to catch processes as they develop.
4. The teacher is not a controlled variable but is a vital part of the classroom environment.
5. Subjective analysis of qualitative data is often of more interest than quantitative analysis (these data involve affective and metacognitive behaviours in addition to cognitive behaviour).
This type of approach, closer to paradigms of naturalistic rather than of traditional scientific research (see Lester, 1985, op. cit.), opens up naturally to a series of problems in terms of methodology:
Yet beyond such open queries, requiring effort and reflection on the part of researchers, a 'naturalistic' approach does underline the general importance of qualitative and long term exploratory studies in which the teacher is considered as a vital part of the task environment.
In the Italian context the figure of a research teacher (as has been previously mentioned), and the fact that a teacher is appointed to the same class for a whole school cycle, make this type of approach particularly natural, and most Italian research studies can be viewed in this way (see Grugnetti, 1990).
Some studies (Boero, 1988; Boero et al., 1989; Boero, 1990; Boero, 1992; Garuti and Boero, 1992) refer to a curriculum project concerning mathematics and the other main subjects taught in the Italian primary school, in which the activity of solving problems in a given context plays a central role in building up mathematical skills. The project consists in a systematic work in suitable 'experience fields' concerning the natural and social reality (for instance: 'productions in the classroom', 'history of the family', 'economical exchanges', the 'sun shadows', and so on). Most of the mathematical problem situations proposed to the children are inserted in these 'experience fields'. However, the work in each experience field also concerns many non-mathematical activities (for instance, performing experiments and writing reports about them; studying historical and geographical topics about the experience fields; and so on). These activities naturally produce many non mathematical problem situations. Particular importance is given in the project to the verbalization processes and to the activities aimed at developing verbal competencies. Regarding the solving of arithmetical problems in particular (Boero, 1988) the project focuses on the development and use of verbal language, and the development of problem solving strategies by means of the resolution of arithmetical problems without numerical data in various fields of experience and through the gradual progression from spontaneous calculation strategies to standardized calculation procedures. Research studies that have been carried out within this project are: the influence of the curricular context on the acquisition of some meanings of arithmetic operations and on the evolution of the pupil's problem solving strategies (Boero, 1988); in particular, pupil's behaviour and conceptual achievements in the transition from informal calculation strategies to a written division algorithm (Boero et al., 1988); the influence of the contextualization of problems in experience fields on strategy choice and solution procedures (Boero, 1992). The results of these studies underline in particular the efficacy of the choice of not introducing algorithms early and directly, but of starting from the (informal) strategies built by the children and orienting them progressively towards more efficient, yet meaningful, strategies. This represents an approach that has been successful for over 90% of the pupils.
Other studies refer to experiments expressly aimed at improving problem solving ability in pupils aged 11 to 14 and, more generally, at developing their ability of substanciation and metacognition (Malara, 1990a; Malara, 1990b; Malara et al., 1992; Malara, 1993a; Malara and Gherpelli, 1994). A fundamental role in such studies, based initially on activities like textual analysis and the verbalization of resolution processes, is played by the type of problems proposed to pupils and by the teaching strategies adopted. The problems proposed, in contrast with standard school problems, are articulate and realistic. Their formulation, in particular, follows the characteristics of natural language, and not of the stereotyped and synthetic language typical of standard problems. Thus the text is longer, and moreover explicit questions are missing and data are missing as well (or seem to be missing). The classroom activity with such problems involves two stages. During the first stage, once the pupil has received the text with the problem, he asks the teacher for the values of those data he considers necessary, and hence he solves the problem. This way of proceeding causes a separation between the processes of comprehension of the situation and of search for solutions on the one hand, and the process of execution on the other. During the second stage the critical analysis of the solutions produced is done collectively through a 'balanced discussion'. Within this project various sub-projects with different aims have been carried out, such as mathematization (Malara et al., 1992) and problem posing (Malara and Gherpelli, 1994). General findings suggest that the teaching strategies adopted favour the development, in students, of cognitive capacity such as arguing and hypothesis reasoning (Malara, 1993a) and of metacognitive capacity such as the awareness of their own knowledge and thought processes (Malara 1990b).
Other studies in the field of developing the problem-solving ability aim at teaching the top-down strategy consisting in the analysis and decomposition of a problem into sub-problems (Iaderosa, 1991; Pellegrino and Iaderosa, 1990; Pellegrino, 1994). In order to overcome the motivation difficulties which emerged when the method was tackled with pencil and paper, the approach to such a strategy was achieved through the use of the language LOGO. The findings from this experiment suggest that the activity described on the one hand helps weaker students to solve problems in an almost computer 'guided' manner, on the other it offers better students an opportunity for reflection and comparison of resolution strategies.
If those studies aimed specifically at learning within the school context are excluded, most of the results of research on problem-solving cannot be immediately used by the teacher. These are often highly specialized results obtained through a control of variables which cannot be proposed within the school environment. Nevertheless, we believe that, beyond the specific results obtained, a great contribution to teaching can be provided by those studies, which, whilst highlighting the complexity of problem-solving, advise teachers on the need for greater attention and sensitivity to the various aspects involved. Thus from research on task variables teachers may generally infer that the difficulty of a problem is not solely determined by its mathematical content, but for instance that the formulation of the text (Grugnetti, 1992), or the choice of context (Zan, 1992), or furthermore the manners or setting in which a problem may be presented (Fischbein and Zan, 1989) can have a determining effect on the results. As emphasized by Boero and Ferrari (1988), the comprehension of 'structural equivalence' of problematic situations is a passage which cannot in the least be taken as granted for the child and seems to occur rather late compared with the child's capacity to flexibly master a given problematic situation.
We give here further details on the above mentioned studies.
The research described in Grugnetti (1992) concerns the role of language comprehension in problem solving and, more precisely, how can the wording of a problem statement affect the student's solution. In an attempt to answer such questions, four problems each with different situations and different levels of language difficulty (i.e. with the presence of adverbs and quantifiers) were submitted to 300 children aged 11. The analysis of the protocols confirmed the hypothesis underlying the research, namely that the adverbs are an obstacle in understanding the statement of a word problem.
The research presented in Zan (1992) aimed at defining a context variable (the 'level' of a problem) and investigating its role in mathematical problem solving. Three school problems in probability were submitted to three groups of 100 children aged between 7 and 8 years. The problems had the same mathematical structure, but they differed in some context variables. The results suggest that the 'level' of a problem influences problem solving behaviour more than the other context variables which were considered.
Fischbein and Zan (1989) have analysed how 360 children aged 9 to 12 behaved when faced by a standard arithmetical problem whithout numerical data. The problem was proposed in two different settings: the first was standard (a verbal formulation), the second (which was made up for the research) allowed to observe the processes and not only the products. The results concerning the two settings are discussed in reference to the choice of data and the resolution strategies.
On the other hand, the results of research on the influence of variables linked to the subject suggest to teachers the need to pay greater attention to some aspects which are traditionally undervalued in teaching, such as the beliefs a student builds up through his own school experience (Micol, 1991; Zan, 1991-'92), or the significant relation between a pupil's beliefs and his problem solving ability (Poli and Zan, 1995), or the attitudes of the problem-solver towards the problem within various and different contexts (D'Amore 1992; D'Amore and Sandri, 1993a).
More specifically, Micol (1991) illustrates an experiment conducted with 84 children in their fourth and fifth year at primary school. The subjects were asked to construct an 'impossible' problem, with the sole limitation of using given numerical data. The protocols produced show that children possess different beliefs about what an impossible problem is. (In D'Amore and Sandri (1993b) a classification of so-called impossible problems is proposed.)
Poli and Zan (1995) present the preliminary findings of a current research investigating the role of beliefs (about self and mathematics) in mathematical problem solving. The findings presented concern the beliefs about mathematical problems of two groups ('good' and 'poor' solvers) of children aged 8 to 10. A questionnaire with multiple choice answers was submitted to the children. The proposed answers were formulated on the grounds of problem beliefs highlighted in a preceding research carried out on 700 primary school children through open ended questions, such as "What's a problem for you?" (Zan, 1991-92). The findings show that good solvers significantly belong to the category of 'structuralists' (in the sense that a mathematical problem is characterized by the need of using mathematical tools), and more generally that good and poor solvers have a significantly different concept of a mathematical problem.
D'Amore (1992) and D'Amore and Sandri (1993a) present the recent findings of their research showing how the solver's attitudes towards the problem vary according to the presentation, the topics involved, the clauses of the didactic contract.
Some research studies on the processes activated during the resolution of problems, however, highlight the variety of paths which may lead to the same product (Bazzini and Grossi, 1988), and the wealth of underlying processes, particularly the role of hypothetical reasoning (Ferrari, 1992; Dettori and Lemut, 1995) and of control processes in the resolution of problems (Gallo, 1994; Tonelli and Zan, 1995). Other studies emphasize the strategy dependence from numerical data (Boero and Shapiro, 1992) and the possibility of getting strategies to evolve by using appropriate contexts (Boero and Garuti, 1992). Such results may suggest that it would be opportune for teachers to pay more attention to factors which could influence resolution processes, and in particular, such results would stimulate them towards having a different approach to errors. The work of D'Amore et al. (D'Amore, 1993b; D'Amore et al., 1995), where the results of the use of natural language in the context of the resolution of problems is discussed, should be interpreted in such a light. A special case is the study of Baldisseri et al. (1994) carried out in an infant-school on how to approach an arithmetical problem at a school level which apparently lacks specific 'didactic contract'í and expectations on the part of the pupils.
More specifically, Bazzini and Grossi (1988) emphasize the importance of analysing problem solving strategies in order to check the acquisition of arithmetical skills in primary school children involved in curricular experimentation. The resolution processes are classified into different categories, and in particular the analysis of the wrong answers is proposed.
In Boero and Garuti (1992) an exploratory study is performed involving six proportionality problems proposed in two classes by the same teacher over a period of about ten months. The problems dealt with different settings (the geometrical setting and subsequently the arithmetical setting). The purpose of the study was to explore the transition to a multiplicative model, the conditions which may enhance it and the difficulties connected with the transfer of a model constructed in the geometrical setting to an arithmetical one. The results show that problem situations considered as such by the students, and relevant to a context in which the geometrical-physical aspect experienced directly is paramount, may be used by the teacher to motivate the student's transition from an additive model to a multiplicative one.
Boero and Shapiro (1992) consider the outcomes of 1023 pupils aged 9 to 13 from two different instructional settings, on a problem involving several variables and solution strategies. The findings from this study led to some interesting interpretations regarding the student's transition to pre-algebraic strategies and the associated mental processes.
Ferrari (1992) analyses the functions of hypothetical reasoning in mathematical problem solving and discusses the interplay between the production of hypothetical reasoning and the comprehension of logical connectives. The investigation was conducted through the analysis of protocols produced by primary school children aged 8 to 10.
Dettori and Lemut (1995) discuss the relations between hypothesis and representation production in problem solving in primary school, viewing a resolution process as a sequence alternating representations and hypotheses.
Gallo (1994) analyses the role of the 'ascending' and 'descending' control in resolution processes activated by subjects aged 14 to 16 when faced by a geometrical construction problem. Such a problem is an 'activity', as is described in Gallo (1988 and 1991). The analysis of the protocols showed that the rigidity of the mental models does not allow the majority of subjects to get a correct answer.
Tonelli and Zan (1995) emphasize the role of metacognitive processes in problem solving. After a brief general introduction the paper deals specifically with strategic decisions and Schoenfeld's related model. The protocol of a first year University student in Mathematics is analysed.
D'Amore (1993b) focuses on the student's difficulty to use the natural language in a general mathematical context and in the problem solving activity: the student often believes that the use of specific and technical language terms is necessary. D'Amore et al. (1995) present an investigation carried out in various countries about the way in which primary school pupils re-interpret the texts of given problems. This study revealed different implicit clauses in the didactic contract concerning the subject 'problems' and the different kinds of difficulties that pupils come up against in forming mental model of the situation described in the text.
Baldisserri et al. (1994) present an investigation carried out with five years old children in infant schools (pre-junior). An arithmetical subtraction problem was proposed to the children. The various ways of approaching the task, which was completely new to the children, was analysed.
The various aspects which have been underlined, the influence of context, or of the verbal formulation of a problem, the importance of a pupil's beliefs, and the variety and wealth of resolution processes, seem to be of particular importance when the teacher finds himself faced with the failure of a subject. Awareness in the role of the various factors involved in the problem-solving activity is necessary to interpret those difficulties which have been highlighted and hence to help the subject to overcome them.
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| Bruno D'Amore | Rosetta Zan |
| Dipartimento di Matematica | Dipartimento di Matematica |
| Piazza di Porta San Donato 5 | Via Buonarroti 2 |
| I - 40126 Bologna - Italy | I - 56127 Pisa - Italy |
| E-mail: damore@dm.unibo.it | E-mail: zan@dm.unipi.it |