Rend. Sem. Mat. Univ. Poi. Torino Voi. 52,2(1994)

WALT1

P. Boero

ABOUT THE ROLE OF ALGEBRAIC LANGUAGE IN MATHEMATICS AND RELATED DIFFICULTIES

Abstract I will try to show how the "transformation fiinction" of the algebraic code enters into action in different mathematical activities; what are the cognitive processes, and especially the prerequisites involved in it; what are the consequences of such analyses on the educational level.

1. Introduction

The aim of this paper is to analyse some epistemological, cognitive and didactical aspects of the "transformation function" of the algebraic code.

The importance of this function in the history of mathematics and in current mathematical activities is relevant: we cannot imagine the development of important chapters of pure and applied mathematics (from theory and applications of differential equations, to linear algebra), without the essential contribution of the "transformation function" of the algebraic code.

From the educational point of view, the "transformation function" of the algebraic code sets many problems. Students do not learn easily to transform algebraic expressions in a standard way: in Italy over one year (almost full time) is devoted to this apprenticeship. The difficulties become more relevant if students must transform algebraic expressions in a non-standard way in order to prove a theorem or to investigate some properties of a mathematical or physical phenomenon, "modelled" by an algebraic expression. At the beginning of university courses in mathematics, I can estimate that more than 40% of students meet with this kind of difficulties (and most of these students are selected through high school courses requiring good mastery of standard algebraic manipulations!).

For these reasons I think it necessary to try to better understand how the "transformation function" of the algebraic code enters into action in different mathematical activities (epistemological analysis, § 2); what are the cognitive processes, and especially the prerequisites involved in it (cognitive analysis, § 3) ; what are the consequences of such analyses on the educational level (educational aspects, § 4).

162 P. Boero

We will see that the "transformation function "of the algebraic code plays different roles in mathematical activities (in relationships with different kinds of problems), and each of these roles implies a specific, different cognitive engagement by students. In such a situation, our working hypothesis for classroom activities is that teachers must equilibrate the activities they propose their students (avoiding an exclusive "imprinting" and a major development for standard transformations), and students must be gradually and directly involved in reflecting upon the variety of roles of the "transformation function" of the algebraic code in mathematics.

2. The transformation of algebraic expressions in mathematical activities

I will use, with heuristic purposes, some diagrams of this kind:

form 1 —• form 2 form 1 form 2

T I I ! sem 1 • > sera 2 sera 1 —> sem 2

In these diagrams, form means any expression based on the use of the algebraic code; this very wide deflnition covers a great deal of mathematical expressions (eventually integrating special symbols used in different mathematical fìelds: mathematical analysis, linear algebra, probability....): from arithmetic expressions (like 3 * [ 2+ 5 * (7+2 * 3)] ) to algebraic equations, from trigonometrie equations (like sin(2x -f 7r/2) + 2cos(x — 7r/2) = 1) to differential equations (like y'\x) = ay(x) — by2(x)), from functional expressions (like f(ax + by) — af(x) +bf(y)) to matrix expressions.

At present, I am not able to give a precise deflnition of sem; due to the heuristic use I will make of these diagrams, it is sufficient to consider sem as a mathematical or non-mathematical cultural object (a mathematical statement, a relationship beetween physical or economical variables, and so on) which may be "put into a formula" (for different, more systematic studies concerning form and sem, see Arzarello, Bazzini & Chiappini (1992, 1993))

In these diagrams, upward arrows mean "formalization", downward arrows mean "interpretation", continuous horizontal arrows mean "transformation" ; more precisely:

- horizontal arrows between form 1 and form 2 mean "transformation according to syntactic rules of the algebraic code", including not only standard algebraic transformations on a literal expression, but also resolution of differential equations, of systems of linear equations, and so on; we will also include substitutions of numerical values to ietters; in general, "transformation" will mean any process, based on direct algebraic transformations or substitutions or general theorems proved through algebraic transformations, and expressed through formulas, which allow to get some new algebraic expressions from the originai one.

About the Role of Algebraic Language 163

Here are some examples:

from: (a4 - 64)/(o + b) to: a3 - a2b + ab2 - b3

through decomposition: a4 - ò4 = (a + 6)(a3 - a2ò + ab2 — b3) and simplification.

from: (sin a; exp2a;)' to: (cosa; 4- 2 sin a;) exp2a; according to the theorem concerning the derivative of a product, and application of the distributive property;

from: x2 4- 2a; — 3 = 0 to: x\ — \,x<i = — 3 through auxiliary, general formulas:

xx - (-6 + \/b2 - 4ac)/2a, a;2 = (-6 - \ /ò 2 - 4oc)/2a

(or longer, ad hoc transformations: x2 -\- 2x — 3 = x2 -i- 2x -\-1 — 4 = (x -\- l ) 2 — 4; so:

(x + l ) 2 = 4 and fìnally a; + l = 2,a; + l : = - 2 which gives x\ = 1, a;2 = —3)

from: ?/'(a;) -f 4?/(a;) = 1 to y(x) = ^4sin2a; + 5cos2a; + 1/4 through standard methods

of resolution of linear differential equations

- horizontal arrows between sem 1 and sem 2 mean "transformation of meaning", in the domain of validity of meanings. I observe that form 1 may be equivalent (through reversible algebraic transformations) to form 2 in the domain of validity of form 1 and form 2, and sem 1 may be equivalent to sem 2.

> will indicate a "guess" (conjecture to be proved, etc.)

2.1. Applying a formula to solve a mathematical or non mathematica} standard problem

In this case, we start from sem 1, we put the problem we must solve into a formula (form 1), we operate a standard algebraic transformation (for instance: solving a standard algebraic equation), and we get a "result" form 2; the interpretation of form 2 produces a new "meaning", sem 2. In many cases, this process is a multi-step process (with a chain of fundamental diagrams of the type considered before).

EXAMPLE 1. It is well known that the "stop space" of a car, from the point where the driver appiies the brakes, depends on the square of the speed; if we want to know what is the "stop space" s from the point when the driver sees the danger, we must add a space proportional to the speed (depending on the quickness of reflex). If we wish to know what speed is possible not to exceed the whole "stop space" of 100 m, we may put the hypothesis stated before (sem 1) into a formula; we gei? as form 1: s — Av2 + Bv < 100; then we may give values to A and B depending on the conditions of the road, on the condition of the braking system of the car, and on quickness of reflex (particularization of the situation, bringing to sem 2).

For a common situation (modem cars, normal conditions of the road, mean reflex

speed) we may pose: A = 0.6; B = 0.08 (if v is expressed in km/h, and 5 in metres).

So, we may solve the inequality: S = 0.006t;2 + 0.08v < 100 (form 2)

164 E Boero

We get (through standard formulas): -136 < v < 123 (form 3) Interpreting this result, we may say that the speed must not exceed 123 km/h (sera 3). The follo wing diagram synthetizes the whole process:

form 1

sera 1 •

->form 2 -T

-> sera 2

->form 3 i

sera 3

We may observe that the root -136 (obtained through the resolution of the equation) is meaningless to our jsroblem; this shows the importance of the "interpretation" phase of the "algebraic result" form 3.

2.2. Producing new knowledge about an open problem situation

The new knowledge may concerni

- the conjecture of the existence of a transformation between form 1 and form 2, suggested by relationships existing between sera 1 and sera 2

A very simple example concerns the evaluation of the area of a rectangular trapeze:

b

formi = ah/2 + bh/2 form 2 = ah 4- (b — a)h/2

In this case, two different decompositions of the originai figure into simpler figures generate two different formulas; but sera 1 is equivalent to sera 2, and this suggests that a transformation may exist between form 1 and form 2

form 1-T

sera 1

>form 2 T

sem 2

We remark that this example may be proposed to comprehensive school students as an introductory activity to consciuos transformation of algebraic formulas.

We may remark also that, in general, the existence of a physical or mathematical, meaningful transformation between sera 1 and sera 2 does not imply the existence of an

About the Rote ofAlgebraic Language 165

algebraic transformation between form 1 and form 2 (where form 1 and form 2 are any two formulas respectively expressing sem 1 and sem 2).

- the existence of an "object" related to sem 1, whose existence is a consequence of the interpretation of form 2, derived from form 1 according to more or less standard transformations A very simple example suggested by Paolo Guidoni is the following:

it is not difficult to verify through measures that the equilibrium temperature Tf reached by the mixture of two quantities of water, mi and m2, is related to their respective temperatures T\ and T2 at the moment of the mixture according to the following formula (form 1):

Tf = (raiTi + m2T2)/(mi • + m2)

This formula may be interpreted as:

"Tf is weighted mean of the temperatures T\ and JT2" (sem 1)

By a very easy algebraic transformation we may write :

Tf(mi + m2) = raiTi + ra2T2

This formula (form2) may be interpreted as "conservation of the quantity of heath" (sem2).

By a suitable algebraic transformation of this formula we may write the following formula:

(Tf-T1)/(T2-Tf) = m2/m1 (form 3)

This formula may be interpreted as "inverse proportionality betweenthe quantities of water and the absolute variations of temperatures" (sem 3). The following diagram synthetizes the whole process:

formi—> form 2—> form 3 T 1 I

semi sem 2 sem 3 Some of the more spectacular applications of mathematics to physics concern this kind of usage of mathematics; physicists "put a physical situation (sem 1) into a formula" (formi) (an algebraic formula, a differential equation, età); suitable (more or less standard) transformations of the formula may generate a new "formula" (form 2). In general, this "formula" may be a very simple expression, like fi = 1 - for instance, concerning an eigenvalue obtained through the resolution of an equation like det(M — p,I) = 0; or a relatively complex formula, like in the previous example; or the equation of an asymptotic solution of a differential equation, the interpretation of which (sem 2) may increase our knowledge of the physical world.

Here it is interesting to remark that a kind of principle of "neutrality" (in relation with the real world) of algebraic transformations (already realised by Galilei) allows to operate in such a way, that if an hypothesis sem 1 is appropriately put into a formula

166 P. Boero

forra 1, the interpretation of a transformed forra 2 formula (obtained from forra 1) may be used as a tool to validate sera 1.

2.3. Proving a conjecture (in mathematics, physics, and so on)

. In this case, frequently, sera 2 is known (the "content" of the conjecture), sera 1 is known (information about data: physical situation, relationships between variables), formi and forra 2 must be expressed in order to get form 2 starting from forni 1 with suitable transformations.

form 1 —> form 2 T i

sem 1 • > sera 2 In many situations, the passage from form 1 to formi (and, consequently, from sera 1 to -

sem 2) needs intermediate steps, according to a chain which may be more or less complex.

As an example of a relatively simple trasformation, we may consider the proof of

the following theorem : "Ifwe consider the homogeneous, first order differential equation: y'(x) = u(x)y(x)

with coefficient u continuous over an interval (a, 6), every solution may be expressed by y(x) = KexpU(x), where U(x) is a primitive function ofa(x) (Le. U'{x) = u(x))".

To prove this theorem, we may consider a solution of the differential equation, say z(x); transforming the nature of the próblem at the formai level, under semantic control (expU(x) is positive on (a,b) ) it is sufficient to prove that z(x)/expU(x) is a Constant function on (a, 6); and to prove it, by a transformation of the problem at the semantic level we may prove that (z(x)/exp U{x))' = 0 on (a, 6).

Then we may perforiti the derivation, according to standard transformation ruìes:

(z(x)/expU(x))' = (z,(x)expU(x) — z(x)(expU(x))')/exp2U(x) =

(zf(x)expU(x) — z(x)u(x)expU(xy)/exp2U(x) =

(anticipation suggests to put expU(x) into evidence)

(z'(x) - u(x)z(x))expU(x)/exp2U(x)

(nere interpretation of the first factor is needed, referred to semantics)

The following diagram synthetizes the whole process:

formi—>form2 formS—> form 4 . T I T I sem 1 —> sem 2 —> sem 3 sem 4

About the Rote of Algebraic Language 167

3. Cognitive aspects

I will consider some research problems concerning prerequisites, approach and tools of algebraic transformations.

3.1. Some cognitive roots ofalgebraic transformations

This subject is under investigation in collaboration with Lora Shapiro.

First of ali we recali some essential contents of our PME-XVI research report (Boero & Shapiro, 1992).

The purpose of this study was to better understand the mental processes ( i.e. planning activities, management of memory ...), underlying students' problem solving strategies in a "complex" situation. Towards this end the following problem was administered:

"With T liras far stamps one may mail a letter weighing no more than M grams. Maria has an envelop weighing E grams; how many drawing sheets, weighing S grams each, may she put in the envelop in order not to surmount (with the envelop) the weight of M grams ?"

Various numerical versions nave been proposed to different classes from grade IV

money maximum weight of the weightof each

needed (T) admissible weight (M) envelop (E) sheet of paper (S)

1500 50 7 8

2000 100 14 16

2000 100 7 8

3800 250 14 16

The students' strategies resolutions hàve been analysed according to a classification scheme suggested by the data from a pilot study, and corresponding to tne aim of exploring mental processes underlying strategies.

Strategies were coded in the following manners:

"Pre-algebraic" strategies (PRE-ALG.). In this category the strategies involved taking the maximum admissible weight and subtract the weight of the envelop from it. The number of sheets is then found multiplying the weight of one sheet and comparing the product with the remaining weight, or dividing the remaining weight by the weight of a sheet of paper, or through mental estimates.If the problem would be represented in

to grade Vili::

(50,7,8)

(100,14,16)

(100,7,8)

(250,14,16)

168 R Boero

algebraic forni, these strategies would correspond to transformations from:

Sx + E<M to : Sx<M-E, up to : x = (M - E)/S

For the purposes of this research, we have adopted the denomination "pre-algebraic" in order to put into evidence two important, strictly connected aspects of algebraic reasoning, namely the trànsformation of the mathematical structure of the problem ("reducing" it to a problem of division by performing a prior subtraction) ; and the discharge of information from memory in order to simplify mental work.

"Envelop and sheets" strategies (ENV& SH).This "situational" denomination was chosen by us because it best represented students' production of a solution where the weight of the envelop and the weight of the sheet are managed together. These strategies include -"mental calculation strategies", in which the result is reached by immediate,simultaneous intuition of the maximum admissible number of sheets with respect to the added weight of the envelop; "trial and error" strategies in which the solution is reached by a succession of numerical trials , keeping into account the results of the preceding trials (for instance, one works on the weight of some number of sheets and adds the weight of the envelope, checking for the compatibility with the maximum allowable weight); "hypothetical strategies", in which one keeps into account the fact that the weight of one sheet is near to the weight of the envelop, and thus hypothesizes that the maximum allowable weight is filled by sheets, and then decreases the number of sheets by one.. and so on.

A preliminary review of the results shows that there is a clear evolution with respect to age X instruction from ENV & SH. strategies towards PRE-ALG.strategies within and between numerical versions (this is found in homogeneous groups of pupils: transition from IV gradò'-to V grade; and from VI grade to Vili grade).

We see that the motivations and access to pre-algebraic strategies may be different; but in ali of them there is a form of reasoning that may derive from a wide experience involving production of "anticipatory thinking". That is to say , under the need of economizing efforts, pupils pian operations which reduce the complexity of mental work. This interpretation provides a coherence amongst different results, concerning the evolution towards PRE-ALG. strategies with respect to age, as shown in the solutions produced in grade IV to grade V and in grade VI to grade Vili, as well as with respect to the results involving more difficult numerical data.

Ali this may also explain different results obtained by Genoa Group Project classes and "traditional" classes: the large experience of subtraction/division problems presented as two steps problems (with an intermediate question) in "traditional" classes does not seem to produce ali the desired effects. Experiencing time separation of tasks, according to the suggestions contained in the text of the problem, may not effectively develop planning

About the Role of Àlgebraic Language 169

skills in the same direction.

Concerning research findings in the domain of pre-algebraic thinking, we may

observe that mere is some coherence between:

- our results, concerning an applied mathematical word problem , proposed to

students prior to any experience of representation of a word problem by an equation and

prior to any instruction in the domain of equations; and

- Herscovics & Linchewski's results, concerning numerical equations proposed to

seventh graders prior to any instruction in the domain of equations. For instance, they

find that an equation like Ari + 17 = 65 is solved performing An = 65 — 17 and then

n = 48 : 4 by 41% of seventh graders, while an equation like 13n + 196 = 391 is solved

in a similar way by 77% of seventh graders. This dependence of strategies on numerical

values is similar to that shown in our tables (compare data concerning sixth graders in the

cases (50,7,8) and (250,14,16) in Boero & Shapiro, 1992).

Filloy & Rojano define (for numerical equations) the "didactic cut" "as the moment

when the child faces for the first time linear equations with occurrence of the unknown on

both sides of the equal sign"; for applied mathematical word problems, a "didactic cut"

might be considered when the child faces for the first time a problem where a separation of

tasks (through an inverse operation) must be performed in order to simplify mental work

and avoid "trial & error" methods . Taking into account the Herscovics & Linchewski's

(1991) and Filloy & Rojano's (1989) findings, we have performed a further analysis of

our data which places into evidence two extreme opposite patterns, and many intermediate

behaviours of pupils engaging in a PRE-ALG.strategy.

Some students seem to transform the problem situation thinking about the number

of sheets and the weight of the envelope as physical variables as they subtract the weight

of the envelope and work with the remaining weight. Other students "put into a numerical

equation" the problem situation (even if they do not formally write the equation!) and

transform the equation (they perform a subtraction, and then a division òn pure numbers).

The presence of these extreme patterns in the same problem situation in the same classes

may explain a deeper relationship between our findings and other findings concerning purely

numerical equations. It also allows us to better understand the degreeto which different

approaches to the "transforming function" of the àlgebraic code are complementary.

In what sense does the study reported before gives information about the cognitive

roots of àlgebraic transformations?

As we saw in the preceding paragraph, àlgebraic transformations (especially the

more open and complex ones) require that the subjet integrate two or more of the following

activities:

— transforming the nature of the problem (through horizontal or vertical arrows), in

170 P. Boero

order to be able to manage the transformed problem in an easier way

•-. anticipating (i.e. imagining the consequences of some choices operated on algebraic

espressions and/or on the variables, and/or through the formalization process)

- . making choices in order to get the solution in an economie way

- suspending the reference meaning (sera) of algebraic expressions, and working at a

purely syntactical level

- using the reference to the meaning (sera) to pian further steps of transformation of

forni, or to interpret the consequences of performed transformations

If we consider the "sheets and envelop" problem and the resolutions achieved by

students, we realize that (depending on age and instruction) many of them were able to

integrate some of these activities in an effective way, choosing different strategies for it.

3.2. Approaching transformations of algebraic expressions

Traditional mathematics instruction drills pupils in "one-way" transformation of

expressions:

- the transformation of numerical expressions is performed almost exclusively to get a

number by applying the rule "calmiate multiplications and divisione, then additions

and subtractions, front the inner parentheses ouf\ and

- the transformation of expressions with letters is performed to get a very simple final

expression by combining distributive property, commutative property, and standard

identities such as (b + a) * (b - a) = b2 - a2.

These prevailing activities might produce an "imprinting" such as only substituting

numbers in literal expressions and computing, or simplification tasks are taken into account

by students. On the contrary, we note that frequently in ordinary mathematical activities

the final result of transformations performed on algebraic expressions may not be simpler

than the originai expressions.

A pilot experiment was performed, concerning four groups each of about 20 students:

(A) : eighth graders with an essential, short, reflective activity, and drill in

the calculation of arithmetic expressions and some introductive notions concerning the

transformation of expressions with letters;

(B): eighth graders with a standard (very extensive!) drill in the calculation of

arithmetic expressions and some introductive notions concerning the transformation of

expressions with letters;

(C) : ninth graders with an essential, short, reflective activity, and drill in the

calculation of arithmetical expressions and in the transformation of expressions with letters;

and

About the Role of Algebraic Language 171

(D) : ninth graders with a standard (very extensive!) drill in the calculation of

arithmetical expressions and in the transformation of expressions with letters.

A and B were eompared in a test requiring to:

1) ascertain whether a formula was equivalent to other formulas or not;

2) simplify standard expressions with letters.

C and D were eompared on a similar test (with more difficult items).Here is an example of the exercises of type 1) for C and D groups:

"Consìder the following formulas: mF + nB — nF + mA; m(F — A) — n(B — F); mF + mA = nB 4- nF; determine the formula equivalent to F = (mA + nB)/(m + n), and perform a suitable transformation from one to another".

We have found out relatively small differences between A and B (perhaps because letters create a new "contract "). We have found very deep differences when comparing C and D as the two tests gave opposite results: group C performed significante better in the first test; group D performed significante better in the second test. This last finding may be interpreted as inhibition of planning skills due to models of behavior suggested by drill in the one-way transformation of expressions with letters. >

3.3. Transformation of algebraic expressions: the role ofwritten representation

A pilot experiment was performed in order to prepare a more systematic study to ascertain the role of writing in transforming expressions with letters.

Fifty-four first-year chemistry and biology university students (coming from different high schools: about half from scientific oriented schools, where extensive mathematical activities are performed, including an extensive drill in transforming algebraic formulas; and half from humanistic or professional schools) were invited, during individuai interviews, to mentally perform the simplification of 18 expressions which were verbally communicated to them. The expressions were of two kinds: the first group consisted in expressions with easy patterns of transformations and simplification (for instance: (b2 — a2)/(b-h a)); while the second group consisted in less obvious expressions (for instance: a2b — a(b + ab)).

In both cases, expressions presented various degrees of complexity concerning: the number of letters (from one to three);levels of parentheses (from zero to two: structural complexity); and length (from three monomials to six). Expressions were read twice, very slowly.

The same students also performed a test with standard, written transformations of more complex expressions in a relatively short time (15 minutes for 15 formulas, each requiring at least two passages).

172 P. Boero

The analysis performed suggests the following hypotheses:

- there is a very dose correlation betwèen the type of high school and the perfomances in written tests;

- there is a low correlation between the results in the written test and in the orai test, especially for the group of expressions where no standard pattern of transformation is evident; and

- the complexity of expressions dramatically influences the results of the orai test (especially concerning the number of variables and structural complexity ).

If these hypotheses are confìrmed by more extensive and systematic studies, it might be interesting to deepen the analysis of the involved phenomena by examining:

- the dependence of the "transformation function" of the algebraic code on its written form; and

- the difficulties in connecting the information in short-term memory with planning tasks (anticipation, etc.)

4. Educational implications

From the educational point of view, we may consider some (more or less) current activities performed (or possible) in school (calculating standard arithmetic expressions, transforming algebraic expressions in order to simplify them, understanding and repeating algebraic proofs, producing and proving conjectures expressed with algebraic formulas, discussing the direction of transformations needed to obtain an algebraic expression with given characteristics, etc). We may evaluate the impact of these activities on the development of an equilibrated mastery of the "transformation function" of the algebraic code.

Calculating standard arithmetic expressions: algebraic notations are managed according to very strict algorithmic rules ("'calcitiate multiplications and divìsions, then additions and subtractions, front the inner parentheses puf*) ; the mechanism is blind, some small changes may create big troubles, as it happens in the calculation of

3 *(2 + 5 *[1+ 4* {1 -6*3}] {2+5 * 6})

We observe that anticipation is not stimulated, nor the application of algebraic properties of operations (distributive property...). And, usually, "conventions" are not distinguished from "properties depending on the meaning of operations": concerning this issue, I may report the experience made, in the school year 1991/92, with a group of 126 primary school teachers, attending a course concerning arithmetics; I asked them

About the Rote of Algebraic Language 173

I) why do we write 4 + 5 * 6 = 34 and not 4 + 5 * 6 = 54 ?

II) why do we write 4*(5 +6) = 4*5 + 4*61

Only 11 teachers were able to write (in a more or less precise way!) that the first equality depended on conventions, while the second depended on properties of operations with naturai numbers!

Transforming algebraic expressions in order to simplify them: in some cases, the final result is given (and it may stimulate some anticipation process, like in an algebraic proof of a given formula); in other cases, anticipation is needed to perform transformations which (for some phases of the process) do not go in the direction of "reducing the number of parentheses": for instance, the simplification of the expression:

(ò 3 -a 3 ) (ò 2 + a 2 ) / ( ò 4 - a 4 )

needs to temporarily add some parentheses, anticipating the fact that ò4 - a4 may liberate b — a and b2 + a2, and that ò3 — a3 may liberate b — a. But most expressions proposed in the textbooks suggest that we go in the sense of progressive simplification, step after step, reducing the number of parentheses, as in this example:

[(b2-a2)(b2 + a2)-b4-a4]/a3.

Under standing and repeating algebraic proofs: this activity is very common from high school on, in algebra (theory of groups, vector spaces, ...) and in other domains while using the algebraic code.No anticipation is requested, while standard patterns of transformation are exerced and may be better understood.

Producing and proving conjectures expressed with algebraic formulas, discussing the direction of transformations needed to obtain an algebraic expression with given characteristics: these are unusual activities in pre-university mathematics education; they might be used to enhanee anticipation and (under a suitable guide by the teacher) to stimulate awareness of the nature of processes of transformation (metacognitive aspect).

By striking a balance between 'common' and 'uncommon' activities perfomed with the algebraic code, we fìnd that the activities more suitable to ensure the development of "anticipation" and a conscious management of the process of transformation are 'uncommon' in school. At present, students are mainly forced to develop the "standard patterns of transformation" component of the transformation process.

REFERENCES

ARZARELLO F., BAZZINI L., CHIAPPINI G., 1992, Intensional semantics as a tool to analyze algebraic thinking, to appear in: Proceedings WALT1, Rend. Sem. Mat. Univ. Poi. Torino 54, 2 (1994), pp. 105-126.

174 R Boero

ARZARELLO F. -BAZZINI. L,, CHIAPPINI G., 1993, Cognitive Processes, In: Algebraic Thinking: towards a theoretical framèworlc, Proceedings PME-XVII, Tsukuba, voi. I, 138-145.

BOERO P., SHAPIRO L., 1992, Ón Some Factors Influencing Studenis' Solutions in Multiple Operations Problems: Results and Interpretations, Proceedings PME-XVI, Durham, 89-96.

CORTES A., VERGNAUD G., KAVAFIAN N., 1990, From arithmetic to algebra: negotiating a jump in the leaming process, Proceedings PME-XIV, Voi. II, Oaxtpec, 27-34.

FiLLOY E., RoJANO T., 1989, Solving equations: the transition from arithmetic to algebra, For the Learningof Mathernatics, voi. 9.2, 19-26.

HERSCOVICS N., LINCHEWSKI L., 1991, Pre-algebraic thinking: range ofequations and informai solution processes used by seventh graders prior to any instruction, Proceedings PME-XV, Assisi, Voi. II, 173-180.

KIERAN C , 1989, The early leaming of algebra: a structural perspective, (Eds. Wagner S. & Kieran C), Research lssues in the Learning and Teaching of Algebra, L.E.A., Hillsdale, 33-56.

LESH R., 1985, Conceptual analyses of mathematical ideas and problem solving processes, Proceedings PME-9, Noordwijkerhout, 73-96.

NESHER P., 1988, Multiplicative school word problems: theoretical approaches and empirical findings, In: Number concepts and operations in the Middle Grades (Eds. J.Hiebert & M.Behr), NCTM, Reston , 141-161.

NESHER P., HERSHKOVITZ S., 1991, Two-step problems - Research findings, Proceedings PME-XV, Assisi, Voi. III, 65-71.

VERGNAUD G., 1988, Multiplicative structures, In: Number concepts and operations in the Middle Grades (Eds. J.Hiebert & M.Behr), NCTM, Reston, 141-161.

Paolo BOERO Dipartimento di Matematica, Università di Genova Via L. Alberti 4, 16132 Genova, Italy.

Lavoro pervenuto in redazione il 29.9.1993