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Studies on Teaching and Learning Algebra

Luciana Bazzini, Elisa Gallo and Enrica Lemut


1. Introduction
 
 

Research studies on algebraic teaching and learning play a key role in mathematics education, as witnessed by the great number of publications and the live debate at international conferences.

 Such studies point out crucial issues in learning mathematics: for example, passing to formalization by means of the use of letters, mastering procedural and structural aspects and the interaction between the use of the computer and the concepts formation.

 Italian research studies on algebra are framed in such a debate; they are rooted in previous research and have recently received new emphasis, which has led, for example, to the organization of the WALT Conference (What is Algebraic Learning, in Turin) in 1992, to a specific session on algebraic thinking at the annual National Seminar (Pisa, 1992), and several kinds of cooperation involving also forein researchers (for example the SFIDA meetings with French colleagues).

 Research studies on algebraic thinking are carried out by several didactic research groups in different places: universities or centers supported by the National Research Council.

 Their contributions concern a wide spectrum of issues and deal with many aspects, different and complementary, at different levels of scolarity.

 Several studies focus their attention on different stages of the students' cognitive development: some of them deal with pre-algebraic thinking, the transition-break from arithmetic to algebra, the mastery of algebraic language, both as object and tool. Other studies focus on learning processes; they are mainly of the following three types:

As far as teaching procedures are concerned, contributions are of a different nature: some of them are proposals for the classroom, others concern specific methods to deal with students' difficulties.

 On the basis of the existing studies, we would like to present the spectrum of Italian research on algebraic thinking, taking into account that some contributions are concerned with more than one aspect, in accordance with the widely mentioned complexity of algebraic teaching-learning problematiques.
 
 

2. Key processes in algebraic thinking
 
 

The general consensus on the importance of algebraic language in mathematics has led some Italian researchers to study key processes which are present in the development of algebraic thinking. Algebraic thinking is considered both as register for representing and as tool for solving problems.

 Processes of "naming" and "transforming" are studied from a general standpoint; processes of "generalizing" and "modelling" are mainly considered as related to problem solving and computer supported teaching. Attention is also given to situations involving the transition from arithmetic to algebra and related students' difficulties. Finally, the role of technology is considered in its potential to help students. As concerns the naming process, special attention is due to the study carried out by Arzarello, Bazzini and Chiappini (1994). Making explicit reference to Frege[1], the authors point out the difference between "sense" and "denotation" of a symbolic expression as a key element of algebraic thinking in problem solving; they investigate this distinction and the mutual interaction of the poles. By "denotation" of a symbolic expression in algebra they mean the number set that is represented by the expression; on the contrary, the "sense" consists of the way the denoted set is given through a symbolic expression. They focus their attention on the relationship between "sense" and "denotation" during the process of assigning names to the elements of a problems. For the authors, the process of naming consists in assigning names to the elements of a problem and is aimed at constructing an algebraic expression able to make explicit the sense of the problem. The choice of names to designate objects is strictly linked with the control of the introduced variables; the difficulty in the naming process depends on the fact that it is very unusual that algebraic formulas are a linear stenography of what is expressed by means of natural language. Evidence is given to the crucial role of the naming process in determining the success of the solution: in fact it is successful insofar as it allows pupils to grasp the algebraic sense of a problem by the "algebraic sense of an expression", which reflects the meaning of the problem by means of a suitable "contextualized sense". By "algebraic sense of an expression" the authors mean the way the algebraic expression incorporates in its signs a computational rule, by which the denoted set can be obtained; by "contextualized sense of an algebraic expression" they mean a mapping between the culture of the knowledge domain under consideration and the syntactical structure of the expression, where its algebraic sense and its denotation are incorporated.

 By "algebraic sense of a problem" the authors mean the clarification of the relationships among the elements of a problem that the problem exibits when it is interpreted according to the rules of algebraic language.
 
 

Additional references to the naming process in algebra are found in Boero (1995) and in Chiappini and Lemut (1991). It is worth noting that the naming process is tightly related to the representation process (see Furinghetti (1995), Gallo (1995), Gallo et al. (to appear).

 The process concerning the transformation of algebraic expressions has been considered by different authors, from different perspectives.

Boero (1994) focuses on the relevance of the transformation function of the algebraic code in the history of mathematics and in current activities: we can not imagine the development of important chapters of pure and applied mathematics, without the essential contribution of the transformation function. On the basis of difficulties students face when they have to transform algebraic expressions in a non-standard way (for instance in order to prove a theorem or to investigate some properties of a mathematical or physical phenomenon, modelled by an algebraic expression) the author discusses how the transformation function enters into action in different mathematical activities (epistemological analysis), what are the cognitive processes and prerequisites involved in it (cognitive analysis), what are the consequences of such analyses on the educational level (educational aspects). He remarks that the different roles played by the transformation function in mathematical activities imply specific and different cognitive engagements by the students.
 
 

The process of transformation within problem solving oriented situations is analysed in Boero (to appear). A crucial aspect of some strategies of problem solving (which might be used to characterise "algebraic" problem solving) is the transformation of the problem in order to better manage it. Such anticipation allows the process of transformation to be directed towards simplification of the task and its resolution. By "anticipation", the author means the mental process through which the subject foresees the final (and/or some intermediate) shape of an algebraic expression useful for solving the problem and the general direction of the transformation needed to get it. The continuous tension between "foreseeing" and "applying", "guessing" and "testing the effectiveness" allows the productive development of the process of algebraic transformation (for a partially different perspective see Gallo (1994)).

 Boero is more concerned with the process of problem transforming rather than with the process of transforming algebraic expressions. In his view, the process of problem transforming may happen without, before and/or after algebraic formalization. He says that when the transformation happens after algebraic formalization, it is frequently based upon the "transformation function" of the algebraic code. In this case, the manipulation of the algebraic expressions extends enormously the range of possibilities of transformation. He agrees with other researchers that during the transformation process, an at least partial "suspension of the original meaning" of the transformed expression may happen. The anticipation allows planning and continuous feed-back; in the case of transformations performed after formalization, anticipation is based on some peculiar properties of the external algebraic representation.

 The author presents an heuristic model for the transformation process, where he avoids the traditional syntax-semantic distinction showing, by examples, its inadequacy. "Form" (any algebraic expression) and "Sem" (a mathematical or non-mathematical cultural object - a mental representation and an external non-algebraic representation) are linked together when writing or reading an algebraic expression; the transformation from "form 1" to "form 2" may happen with or without making different kind of references to corresponding "sem 1" and "sem 2". The author analyses some cognitive aspects of algebraic transformations, namely the process of anticipation and the role of external representation in it; finally he discusses some educational implications and shows that by the mean of transformation process new knowledge can be produced (about an open problem situation; conjecturing and proving conjectures,...). This last point is also discussed in Chiappini and Lemut ( 1991).

 Gallo (1995) points out that formal transformation of expressions make sense when they are inserted in a conceptually structured context, even if they operate on the ostensive plane of the expression itself. The question at the basis of her research on algebraic manipulation is therefore "what are the dynamics of control during the solution of an algebraic exercise". She says that the adaptation of the models activated by the pupil during the algebraic manipulation, in order to create a model ad hoc, is provoked by the action which links the subjective pole (what the pupil knows) to the objective pole (what the pupil produces), which she refers to as "control"; the manipulation which takes place in terms of formal ostension must also be carried to the conceptual level, with the action of control.

 Starting from a school reality in which in the first two years of high school students essentially "do" algebraic calculation of ever increasing difficulty, and then in the subsequent three years they are asked to apply algebraic rules of manipulation, Menghini (1994) focuses on the different kinds of "control" of algebraic operations that could be acquired by the students (checking by substitution, recognizing known "forms", or mastering and recognizing the rule of syntax). The author prefers the student to get into the habit of using the last two methods of control; she thinks that it is better, at a certain level, to underline explicitly the "leap" from arithmetical algebra to symbolic algebra, giving a major attention to "form". She takes a step back into the past, to the time when the basis for modern algebra was being laid, and remarks that it is worthwhile looking at the epistemological obstacle from an educational point of view and remarks that Peacock underlined the two separate aspects of algebra, distinguishing between "independent" science on the one hand and "instrumental" science for discovery and investigation on the other.

 Reggiani (1994) and, partially, Chiappini and Lemut (1991) , are concerned with the ability to solve a problem translating it into a formula. There is evidence that the mastery of the syntactical rules not always corresponds to the awareness of the generality of what has being done. Based on experimental studies concerning the "spontaneous" ability to pass from particular to general, Reggiani observes that even if in some problem solving situations pupils are able to reach a spontaneous awareness of the generality discovered, the attained level of generalization can not be considered as a stable achievement.

 Finally Cannizzaro and Celentano (to appear) investigate the relationship between Arithmetic Algebra and Symbolic Algebra by the analysis of a test administered to students in upper secondary school. Their data point out the educational relevance of "complete" processes which include formalization, proof and explanation of the results students get in solving problems.
 
 

3. Interpretative models of algebraic thinking
 
 

There is general consensus that the symbolic language of Algebra is a powerful means of approaching and solving problems. It plays a crucial role in increasing the learner's potential for thinking, reasoning and communicating. However, there is strong evidence of students' difficulties in relating symbolic expressions to their meaning; this leads to a more detailed analysis of the meaning of symbolic expressions, together with an investigation of related cognitive processes.

 Arzarello, Bazzini and Chiappini (1994) have given a theoretical frame to understand algebraic thinking in problem solving. They observe that in mathematics there are expressions with the same "denotation" but a different "sense": here "denotation" is concerned with the "extensional" aspects of the expression and "sense" with the "intensional" ones. These words are used according to the Frege distinction between Sinn (sense) and Bedeutung (reference) of an expression.

 For instance, the expressions 4x+2 and 2(2x+1) express two different procedures but denote the same function. Natural language and mathematical language are both rich in expressions with different sense and the same denotation. Algebraic transformations produce different expressions which can have different sense and the same denotation; on the other hand, given two different expressions with the same denotation, it is not always possible to transform one in the other by means of algebraic transformations. This fact is often neglected by students, who recognize a rigid one-to-one correspondence between sense and denotation: sense and denotation are identified. For those students, a symbolic expression denotes only itself: algebra becomes a pure syntax.

 The capacity of mastering sense and denotation is fundamental in algebraic thinking: however, the interplay of these two components give us only flashes and not the entire movie of the cognitive process in action. The analysis is completed by introducing the notion of frame, i.e. a structure of data able to represent generic concepts or stereotyped situations stored in memory. In Semiotics, understanding a given text means activating a frame, making it actual in accordance with the text's values and infering information accordingly. In our context, frames are activated as virtual texts while interpreting a text (for example the text of a problem), and allow to grasp inner relationships according to the context and circumstances expressed by the text itself.

 Thus, to interpret algebraic processes it is necessary to take into account:

From this point of view, doing algebra is a process of giving sense to denotation and denotation to sense and of changing senses and denotations.

 During such processes, the student interprets texts and his/her textual cooperation is fundamental in activating one or more frames. A chain of interpreters is generated: each text interprets the preceding text and, at the same time, it is interpreted by the text which follows in the chain. In each step a change of frame can occur, i.e. a change in interpreting a given expression. Doing algebra appears to be a game of interpretation.

 From an educational point of view, such chain of interpreters, which can be built as the result of the game of interpretation, is very fruitful for the study of learning algebra.
 
 

4. Algebra and problem solving
 
 

The relationships between algebra and problem solving is approached from different perspectives. Algebra can be seen as language apt to represent the problem (or the problem pattern), or as the very core of the problem.

 Boero (1994, 1995) points out the role of algebraic language in mathematics and analyses how the transformation function of the algebraic code enters into action in different mathematical activities, what are the cognitive processes, and especially the prerequisites involved, what are the consequences of such analysis on the educational level.

 Transforming algebraic expressions is framed in the more general perspective of transforming the problem in order to better manage it. A crucial aspect of some problem solving strategies is the transformation function of the algebraic code; it plays different roles in mathematical activities, according to different kinds of problems, and each role implies a specific engagement by students.

 Algebraic transformations, especially the more open and complex ones, require that the subject integrates two or more of the following activities:

Analysing the role of algebraic transformations in problem solving, Boero speaks of dialectics: when a subject has the necessary prerequisites and experience to cope with a problem in need of algebraic transformations, his success depends on a functional dynamic relationship between two poles (standard pattern of transformation and anticipation), whose characteristics are different and, in some sense, opposite. The continuous tension between foreseeing and applying, guessing and testing the effectiveness allows the productive development of the process of algebraic transformation.

On the educational level, the teacher should avoid an exclusive imprinting and a major development for standard transformations; the student should be gradually involved in reflecting upon the variety of roles of the transformation function.
 
 

Chiappini and Lemut (1991) analyse the construction and interpretation of algebraic models in solving problems; their study is mainly concerned with two types of problem situations that may be used to introduce algebra and algebraic models. The first kind, which is very common in teaching, is characterised a priori by the presence of elements that may easily lead the pupil to construct and interpret algebraic formulas by means of shorthand process, not being required to transform the formulas straightway. In these cases, the algebraic model comprises one or more formulas in which letters may be subsequently substituted by particular numerical values.

 The second type is characterised by fewer stimuli towards formula construction and by the intrinsic need to transform the formulas produced. It comes down to writing an algebraic model, which is then manipulated according to the rules of algebraic calculations in the view of solving a given problem.

Such study is framed in the context of a curricular project on teaching mathematics and sciences in comprehensive schools, which also involve the use of the computer. The difficulties students encounter in constructing, interpreting and recalling algebraic models appear to constitute one of the impediments to the transposition of solution strategies into statement sequences and to the interpretation of given statements.

In the analysis of abilities and cognitive attitudes which intervene in the construction and interpretation of algebraic models, the authors underline the following abilities:

Finally the authors state that the problem situations of the first kind seem to be much more accessible for the pupils, but nonetheless they often present the risk of an eccessive emphasis on the shorthand function. In any case, the models produced by the pupils are the working basis to develop the transformation function.
 
 

The transition from arithmetic to algebraic thinking by using a spreadsheet is analysed by Dettori, Garuti and Lemut (to appear). They shift their focus from the potentialities of the software to the main characteristics of the taught subject, and stress that, from an educational point of view, it is not important to learn to find a numerical solution to algebraic problems but rather to understand the nature and power of the theoretical solving scheme of algebra. In such a perspective, it is useful to introduce algebra both in a pen and paper environment and in a computerised one. The research study starts from simple problems where the algebraic solution is close to the arithmetic one and which can be solved by one equation in one unknown. Then, more complex situations, requiring the solution of more equations or inequalities in more unknowns, are approached.

 The authors conclude that using a spreadsheet, which by itself would lead students to solve problems by trial and error, under the attentive guidance of the teacher, can lead them to become aware of activating a new modelling process for problem solution, to understand what it means to solve an equation, even before being able to handle equations, to reason about the domain and constraints of a problem in order to decrease the number of trials necessary to reach a solution and finally to become aware of the nature of algebra by comparing, as a metacognitive activity, different problem solving methodologies, such as arithmetic, algebra and spreadsheet.
 
 

In the research studies by Gallo et alii (1994, to appear), standard algebra exercices are considered as problems to be solved: the analysis is centred on algebraic manipulation occuring during the solution of algebraic exercices, submitted to students aged 14-16, and aimed at the acquisition of the so called "symbol sense" [2].

Two kinds of exercises are submitted to pupils:

Solving such exercises, the necessity of control is evident. It links the subjective pole (what the pupil knows) to the objective pole (what the pupil produces). Control induces the adaptation of the models activated by the pupil during the problem solving process in order to create a model which suits the situation, namely an ad hoc model for the problem to be solved.

 This research study has pointed out that an algebra exercise may be taken as a problem or not. The latter is true when the pupil automatically applies a procedure without using control. If, on the contrary, control is present in the solution, the exercise becomes a problem and control is developed at the level of perception and perceptive reorganization of formulas, at the level of resources and procedures, at the level of syntax and meaning, namely in global and local semantics.

 Control, which appears when the exercise is assimilated, a strategy is elaborated, a choice of model is acted upon and the solution is verified, is caused by the interaction between information/models/products, the conflicts between different models, the rigidity of standard models and the non appropriateness or the incomplete formation of the current model and it produces the evolution (adjustment/formation) of models.

 When control is activated, it is manifested by descendant and ascendant dynamics, i.e. descending

from the mental model to the actions which produce the temporary solution and ascending from the solution produced to a new mental model.

 It can be seen that under the action of descendant and ascendant controls, the solution of an algebraic exercise as problem solving process appears to be a series of open cycles, each of which starts with a representation of the exercise and leads to a new representation.
 
 

Special mention should be given to the so called pre-algebraic thinking, i.e. on the one hand thinking algebraically within a numerical environment or thinking arithmetically within a "lettered" environment and, on the other, building preconcepts relevant and necessary for the construction of formal concepts[3].

As far as the first meaning of pre-algebra is concerned, the studies by Arzarello (1989) and Boero (1994) are worth noting. In Arzarello (1989) the notion of conceptual model is used to study the activity of verbal problem solution in pupils aged 7-13 when they face additive and multiplicative problems. Problem solving is described dynamically as a process of transformation from some basic intuitive models, integrated with the pupil's culture and expressed, by means of natural language, into more elaborated and formal ones. Such models are of an algebraic nature even if they are referred to a numerical context.

 Boero (1994) uses the term "pre-algebraic" to name solution strategies which are framed in numerical context but present some features strictly related to algebraic thinking, namely the transformation of the problem mathematical structure and the discharge of information from memory in order to semplify mental work.

 Studies by Gallo et al. (1988) are concerned with the second meaning of pre-algebra. Such studies analyse solution strategies by elementary school children in playing situations involving the use of numbers with sign and in situations whose algebraic representation is the linear equation by Diophantus. In both cases an evolution of the problem representation is evident, i.e. from reality oriented representations to more formal ones.
 
 

5. Obstacles and difficulties
 
 

Some studies have pointed out difficulties students encounter in learning algebra.

As already mentioned, Boero (1994, 1995) has deepened the idea that transforming the problem is a crucial aspect in some solving strategies. In this perspective, anticipation allows the transformation process to be directed towards simplification and resolution.

 Gallo and Grange (1988) have studied the symbol control moment, i.e. when the student has to face the very meaning of the abstract objects (s)he has to deal with. With reference to general semiotics, which is concerned with the symbols forming and functioning, the authors observe that there are different image levels to represent the world. The adequacy between representation and sensitive perception is expressed by the use of signs, which are arbitrary indexes. The worst inadequacy occurs in the use of signs not related to meanings, as shown in analysing student's behavior in middle school (age 11-14).

 At the same school level it is worth considering a study by Reggiani (1994). Starting from the basic consideration that the difficulties the students have in using algebraic language originate from obstacles of a prealgebric nature, the attention has been focused on the determination of nodal moments in the study of arithmetic and in the passage from arithmetic to algebra. To avoid misunderstandings and lead students to a good use of algebraic formalism, the author points out the awareness of the conventions used in the field of arithmetic and algebra. There is evidence that pupils see algebra as a set of conventions, all at the same levels, difficult to be distinguished and handled.
 
 

At the secondary school level, some studies have analysed standard errors which are rooted in previous misunderstandings and gaps between procedures and meanings.

 Chiarugi, Fracassina and Furinghetti (1990) are concerned with learning difficulties behind the notion of absolute value. On the ground of their teaching experience and research on learning, they point out that the notion of absolute value, while it does not present relevant difficulties when used on numbers, originates errors and misconceptions when used on letters. In order to carry out a longitudinal study on these errors, a questionnaire was administered to students of three different school levels (ages 14, 17,19). The results showed the presence of wrong beliefs on the absolute value, which are difficult to eradicate.
 
 

Gallo et al. (to appear) point out conceptual and procedural aspects of algebraic manipulation. When a student has to cope with an algebraic problem, he has initially to be able to write down algebraic formulas deriving from the problem, but he has also to be able to transform them into other forms, able to provide useful information.

 For many students solving exercises of algebraic manipulation is often a routine procedure, when they use inadequate models, which are influenced by the subject treated at the moment. As a consequence, they use few instruments of control on procedures and results.

Starting from these considerations, the authors have studied situations aimed at investigating and indentifying specific aspects concerned with the meanings students give to algebraic exercises, with motivations leading to solutions and control moments while solving.

 There is evidence that algebraic manipulation requires the subject to operate at three levels, i.e. the objectives level, the resources level and the techniques level. The student is successful when he is able to act at the three levels: otherwise errors occur.

 The contribution by Furinghetti and Paola (1994) is an analysis of the meaning of parameters, unknowns and variables in view of teaching. A questionnaire to establish how students perceive the differences between parameters, unknowns and variables was submitted to students aged 16-17.

 Questions related to the solution of equations and inequalities in upper secondary school is approached by Bazzini (to appear).

 With reference to the interpretative model described in Arzarello, Bazzini and Chiappini (1994), Bazzini points out that in the case of equivalent but not algebraically transformable equations (inequalities), many students do not recognize the invariance of denotation with respect to sense, independently from giving a correct definition of equivalence from a theoretical point of view. In such cases, the non transformability of the algebraic sense inhibits the capacity of recognizing the denotation invariance. In addition, transforming an equation in another one by means of algebraic operations, seems to be a necessary condition for the existence of equivalence.

These results are in accordance with those by Linchevski and Sfard[4], even if the motivation is different. In fact Linchevski and Sfard attribute the cause of such pathologic behavior to the didactical methods which give more emphasis to relational aspects rather than to the procedural ones in the concepts introduction.

6. Implications for teaching
 
 

Some of the papers we have taken into account include implications for teaching. A study by Furinghetti (1995) is concerned with the use and abuse of mathematical symbols in school practice. She observes that students' spontaneous symbols are not encouraged in school to develop towards generally accepted symbols. Such symbols are often the cause of learning difficulties, because of their loss of meaning. The contribution by Malara (1989) deals with the teaching of algebraic structures in the first two years of secondary school and is concerned with the introduction of algebraic language in compulsory school. Firstly some motivations to teaching algebra are pointed out and activities oriented towards structural analogies are proposed. Then a didactic itinerary similar to the learning of natural language is sketched. Other general considerations on teaching algebra in the first two years of secondary school are present in Gallo (1990): the focus is on the introduction of letters as variables and as unknowns. In addition proposals for the passage from numbers to variables and for the transformations involved in the equivalence of equations are given.

 The study by Gherpelli and Malara (1994) is concerned with specific proposals for lower secondary school aimed at considering numbers (natural, relative, decimal and rational) as mathematical objects The authors analyse the passage from natural language to symbolic language in a series of problem situations in which the pupil has to identify properties and regularities, discuss properties under given hypotesis, produce counter examples and proofs.

 Reggiani (1994) presents an experience regarding the construction of some entrance tests to upper secondary school on algebraic topics and the use of relative protocols as tools for diagnosis and as a basis for planning a didactic activity. She refers in particular to the topic of equations where there is evidence of the main difficulties and of the fracture between syntactical rules and the internal semantics of operations. The experience is an example of a working method (i.e. construction of a test, analysis and classification of the protocols, didactic intervention, assessment of the results) that allows the utilizing of the moment of assessment as a starting point for learning improvement.

 A study by Bovio, Reggiani and Vercesi (1995) deals with some didactic problems related to first degree equations in the first two years of secondary school. They focus on nodal points which are not always clear in textbooks and which teachers do not always make as clear as they should. The topics relevant to equations which are dealt with, are the traditional ones, but the purpose is to make them become basic points in didactic planning and not only instruments as they are usually considered.

 Finally Delfrate et al. (1992) are interested in monomials: their approach is different from the traditional one, because algebraic language is treated exclusively as formal language with a specific alphabet and syntax.
 
 

References
 
 

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Luciana Bazzini  Elisa Gallo  Enrica Lemut
Dipartimento di  Dipartimento di  Istituto per la
Matematica  Matematica  Matematica
Università di Pavia  Università di Torino  Applicata - C. N. R.
Via Abbiategrasso, 215  Via Carlo Alberto, 10  Via De Marini 6
27100 Pavia, Italy  0123 Torino, Italy  16149 Genova, Italy

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