Perspectives on School Algebra

PERSPECTIVES ON SCHOOL ALGEBRA

Mathematics Education Library VOLUME 22 !"#"$%#$ &'%()* A.J. Bishop, !)#"+, -#%./*+%(01 !/23)4*#/1 54+(*"2%"

&'%()*%"2 6)"*' H. Bauersfeld, 6%/2/7/2'1 8/*9"#0 J. Kilpatrick, 5(,/#+1 -:;:5: G. Leder, !/23)4*#/1 54+(*"2%" S . Turnau,
=--E(LL555M$345!*#,3+,!M7#: =--E(LL555M!;##$<M$345!*#,3+,!M7#:

TABLE OF CONTENTS Introduction M)+"94#' ;4(,/*2"#'

vii

Approaches to Algebra M)942) E"9D)+ N%#+1 C/*/+" M)O"#)1 52"# 6/22 "#'@M)+"94#'@;4(,/*2"#'

1

The Historical Origins of Algebraic Thinking N4%+ 8: M"'7)*'

13

The Production of Meaning for 52$/3*"P a Perspective Based on a Theoretical Model of Semantic Fields M)942) E"9D)+ N%#+

37

A Model for Analysing Algebraic Processes of Thinking A/*'%#"#') 5*H"*/22)1 N4B%"#" 6"HH%#% "#' 8%"9D")2) E,%"DD%#%

61

The Structural Algebra Option Revisited L".%' / can deduce that x =

the scribe actually assumes a false length, 1 3 = = . He calculates the –

=

=

x0 =

5

(because

. d . In contrast,

which gives a false width false diagonal using

= I ) . The proportional argument under-

lying the procedure leads the scribe to calculate the inverse of arid to multiply this inverse by the given diagonal d=40. This problem clearly shows the functioning of the ancient false position method and will suffice to elaborate our historical reconstruction of the transition from arithmetic to algebra in Mesopotamian land. However, before going on to our next

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L.G. RADFORD

stop in our historical journey, we need to make the following cultural and epistemological remark: the idea of using a 'false quantity' to start the false position method. leans on a deeper and more complex idea: at the beginning of the problem. the 'true quantity' (i.e. the exact solution of the problem) may be legitimately thought of (,*)4$, another quantity. 'False quantities' thus appear as 9/("D,)*+ of 'true quantities'. Furthermore, this is not a phenomenon restricted solely to mathematics: Mesopotamian thinking is full of metaphors. Odes. epic poems, literary and religious texts, for instance, show an intricate system of metaphorical expressions (see e.g. Wilson. 1901). Algebra, we shall suggest. was couched in such a system. ALGEBRAIC THINKING AS A METAPHOR OF THE FALSE POSITION METHOD As we shall see in later sections of this chapter, where we focus on some technical details, the influence of false position methods in the emergence of algebraic ideas can be discerned through some important structural similarities between false position reasoning and early algebraic thinking. One of the studies of the ancient connection between the Babylonian false position method and algebra was made by François Thurcau-Dangin (1938a). Following the trends of the old interpretation of Babylonian mathematics based on the possibility of translating the calculations shown in many of the tablets into modern algebraic symbolism, he noted a strong parallelism between the calculations done in some problems solved by false position methods and those of the modern algebraic methods6. He then claimed that, indeed, some Babylonian procedures were algebraic procedures. Thureau-Dangin’s main idea was supposedly supported by the fact that, in some problems, the scribe takes the number )#/ as the false solution (such an example could be the problem discussed at the end of the previous section) and when, according to Babylonian procedures. we replace the number )#/ by our modern unknown ‘x’, the problemsolving procedures look much like the modern algebraic procedures. He, (as well as others. e.g. Vogel, 1960), claimed that the number one was actually taken as a */D*/+/#("(%)# of the unknown and i f we cannot. straight out, see the unknown, it is simply because the scribes did not have a symbol with which they could represent it. However, the idea that Babylonians developed an _invisible algebraic language’ (i.e. a genuine algebraic language without symbols) has been abandoned (see Radford. 1996a. pp. 39-40). Effectively, there is no clear and safe argument supporting the statement that the Babylonian scribes actually (,)4$,( that the number one */D*/+/#(/' the unknown in an algebraic sense (see Høyrup, 1993b, p. 260). On the contrary. this peculiar numerical choice for the unknown seems to have allowed the scribes to +0+(/9"(%+/ the numerical problem-solving methods and hence to reach an important step in the conceptual development of ancient proportional thinking. In

THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING

17

fact, when ancient problem-solving procedures may be safely identified as «algebraic», which is the case of the problem mentioned in footnote 10, the unknowns are not */D*/+/#(/' by the number ‘1’ or anything else for that matter; instead, the unknowns bear their contextual name (e. g. the length and the width of a rectangle). Rather, the "2$/3*"%B %'/" )7 4#=#)># seems to have been thought of as a 9/("I D,)* of the ‘false quantities’ used in the ancient false position method. It happened, we suggest, when scribes stopped thinking in terms of 7"2+/ b4"#(%(%/+ upon which the false position method is based and. looking at the false quantity metaphorically, began to think in terms of the sought-after object %(+/271 accepting to consider this object as a #493/* (i.e. an operable, inanageable number) regardless that it was not c yet known . This could happen in solving non-practical problems previously solved by false position proportional methods like the following (Tablet YBC 4652, No. 7), where the method of solution is unfortunately omitted: ‘I found a stone, but I did not weigh it; after I added one-seventh and added one eleventh (of the weight and its one seventh). I weighed it: 1 ma-na. What was the original weight? The origin(a1 weight) of the stone was ma-na, 8 gin, (and) 22 1 še ‘ (Based on Neugebauer and Sachs'reconstruction; 1945, p. 101). 8

In modern notations, this problem reads as follows : x +

–

+

–

x+

–

=

1.

The algebraic way of thinking could have even been conceived when ancient scribes 9

faced an even simpler problem. For example, a problem of this type : x + 1 11

–

x = 1.

Let us suppose that this problem concerns the weight of a stone. The false position method is as follows: we assume (according to the usual line of thought in Babylonian mathematics) that the sought-after quantity is 11; then, the stone and one eleventh of its weight is 12. However, we should have 1. This means that we need to reduce the ‘false position’, that is the false value that we assumed at the beginning (i. e. 11). To reduce it, an elementary proportional argument shows that we need to take one 12th of our initial assumption, so the answer to the problem is 11/12 (or 55/60 = 55_ in the sexagesimal system). To see the metaphor that we are suggesting at work, let us, instead of beginning by assuming a 7"2+/ D)+%(%)# or false solution. start the problem-solving procedure by reasoning on the exact unknown sought-after quantity. In this case, the calculations unfold in a different way: first, we multiply both sides of the ‘equation’ by 11 thereby transforming the ‘equation’ into an ‘equation’ without fractional parts. This leads us to an equation that we would write as 12x = 11. Now, following a recurring Babylonian procedure, we just need to find the inverse of 12, which is 5 _ , and to multiply this inverse by 11, which gives us the answer 55_. (Note that the procedure

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L.G. RADFORD

of multiplying both sides of the ‘equation’ by a number is attested to in many Baby10 lonian problem, e. g. C/K(/+ !"(,d9"(%b4/+ '/ Suse ). The type of problems that we have just discussed were frequent in ancient civilisations. For instance, one problem of this type is found in the Egyptian Rhind’s Papyrus; another is found in Babylonian tablets. This is the case of problem No. 3 of tablet YBC 4669 which could be translated into modern notations as follows11: 2 1.x=7. [ [@ @

The conceptual connection between false position ideas and algebraic ones can also be found in post-Greek mathematics. It can be retraced to some mediaeval mathematical treatises. It is particularly enlightening that, in the false position methods, mathematicians. at the beginning of the problem-solving procedure, used to refer to the action of choosing the false numbers as ‘making a position’. In the same way, when a problem is solved by algebra, the introduction of the unknown is sometimes referred to as ‘making a position’. For instance, in Filippo Calandri’s -#" *"B)2(" '% *"$%)#% (15th century). problem 18 deals with a problem that we may translate into modern notations as:

X

= x 1 x+1 (‘Trouva U numero, che partito per uno. più ne vengha un meno’. Santini (ed.), 1982, p. 19). Solving this problem through algebra and by calling the unknown (,/ (,%#$ (‘la cosa’, according to the Italian mediaeval tradition) Calandri says: ‘Farai posizione che que1 numero sia una co(sa)’ (I will make a position so that the (sought-after) number is a thing). An early example ( 14th century) is found in problem 6 of Mazzinghi’s C*"(("() '% A%)**((%: In this problem Mazzinghi says: ‘El primo (modo) e che si faccia positione che lla prima parte sia 5 et una chosa’ (Arrighi (ed.). 1967, p. 23). The connection between false position ideas and algebraic ones is more explicit in an anonymous abacus treatise of the 14th Century: F2 (*"(("() 'e"2$%3*": In this treatise, the unknown is defined just as a position: ‘...in prima noi diremo che sia questa chosa, onde dirò che non sia altro se non f una posizione che si fa in molte questioni...’ (first of all we shall say what this thing is, where I shall say that it is no more than a position that one makes in many questions; Franci and Pancanti, eds. 1988, p. 3, my translation). We can go one step further in our connections between algebraic and false position ideas by referring to a book written in 1522: Francesco Ghaligai’s ?*"((%B" '_5*%(,9/(%B": However. in this case. algebraic ideas have been developed enough to be taken as the explanatory substratum in which the false position ideas are set up. Ghaligai says: —

–

‘We can notice that the position is a concept assimilated to the thing that is chosen according to the knowledge of the intellectual realm Speaking in the case of a thing not known to you, right away the mind will think as if it already knew and say: position is a

THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING

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quantity placed according to the case (the problem) and even though there are two positions, sometimes with only one position the case can be solved and one finds that which is necessary.’ (Ghaligai. 1538. p. 62: my translation).

Ghaligail’s ?*"((%B" 'e5*%(,9/(%B" shows then that the conceptual development of the unknown has completed a loop, now changing the roles of ideas: at that time, the metaphorical-analogical process was reverted and one explains the false position ideas in terms of algebraic ones. methods have to arise, thereby making it possible to handle the algebraic unknown. The metaphorical-analogical process underlying the passage from arithmetic to algebra will map or %#'4B/1 as is the case in inany metaphors (see Lakoff and Núñez, in To end this section let us stress the fact that the new algebraic object of unknown does not come to life alone: it emerges along with new methods. False position methods deal with numbers only. So, new print), important structural features of the first domain here the arithmetical one _ into the second domain here the algebraic one. This metaphorical induction is very clear in inany passages of Diophantus' 5*%(,9/(%B": Indeed, in any of Diophantus' algebraic methods arc hardly understandable without linking them to the ancient false position methods, as we will see in the next section. In order to understand this specific aspect of problem-solving methods. we now need to examine the place of Diophantus' 5*%(,9/(%B" in the development of algebraic ideas. –

–

ALGEBRAIC IDEAS IN DIOPHANTUS' ARITHMETICA

As we know, Diophantus' 5*%(,9/(%B" (written B%*B" 250), is made up of 13 books (3 of them are lost) dealing with the resolution of problems about numbers. Book 1 contains a short introduction in which a division of numbers into +D/B%/+ or B"(/$)*%/+ is presented: the squares, the cubes, the square-squares. the square-cubes. the cube-cubes. Each category contains the numbers that share a similar form or shape (the same /%')+T: Instead of being merely riddles like the Babylonian nonpractical problems, the problems of the 5*%(,9/(%B" were formulated in terms of the mentioned +D/B%/+: For instance, problem 10 from Book 2 reads as follows: ‘To find two square numbers having a given difference’ (Heath. 1910, p. 146). Undoubtedly, within the philosophical principles of Classical Greek thinking (where the search of origins and rational organisation was a starting point), Babylonian numerical word-problems and all the subsequent numerical activity surrounding similar problems in the post-classic Greek period. an activity particularly attested to by the 5#(,)2)$%" $*"/B" (Paton, (ed.), 1979), did not find a suitable niche to prosper 12. By transforming the Babylonian numerical word-problems into problems about abstract Greek species and other ancient well-known riddles that Diophantus supposedly disguised in abstract terms in his 5*%(,9/(%B" (e.g. Book I, problems 16-

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L.G. RADFORD 13

21). Diophantus elevated this unscientific discipline to a scientific one . This was not the only important new aspect incorporated by Greek algebraists. There is another one related to the introduction of %#'/(/*9%#"(/ #493/*+ to the mathematical realm. This was done through a new use of the concept of "*%(,9)+ (ariqmoV), that is, the g#493/*e: ‘The arithmos’, says Diophantus, ‘is an indeterminate multitude of units’ (cf. Ver Eeck, 1926, p. 2) – although, in the solution of many problems, it can be an indeterminate multitude of fractional parts. The subtle, yet fundamental, step made by Diophantus in introducing %#'/(/*9%I #"(/ #493/*+ to the mathematical realm can be better appreciated if we see it within the heritage of the ancient philosophers. In this line of thought, it would be worthwhile to remember that. in one of the few extant fragments of the first Pythagoreans, Philolaus says: ‘Actually, everything that can be known has a Number; for it is im,14 possible to grasp anything with the mind or to recognise it without this SJ493/*T ; and here, Number means a determinate multitude. Thus, by introducing the "*%(,9)+ as an indeterminate multitude Diophantus extended the borders of what can be known. By the same token, the aforementioned concept of arithmos gave way to the creation of a completely new theoretical calculation on %#'/(/*9%#"(/ amount of units (e.g., in modern notations. rules dealing with calculations like x x x2 = x3 or x x 4 = x3) that proved to be very powerful in the resolution of problems. It is important to note that these mathematical accomplishments at the end of the Antiquity were linked to an increasing (albeit not complete) abandonment of Greek classical principles and the spreading of neo-Platonistic speculations that made it possible to think in new, different and promising ways (see Lizcano, 1993). In most of the problems of the 5*%(,9/(%B" the formal structure of their statement follows the same pattern: the problem is stated in terms of operations performed on some categories of numbers (the square numbers in the previous example). At the beginning of Book 1, Diophantus gives a description of the heuristics that one should follow in order to solve the problems. This is the very first explicit ancient description about how to solve problems on numbers that we know. For our discussion we will quote the following extract: ‘...if a problem leads to an equation in which certain terms are equal to terms of the same species (eidos) but with different coefficients, it will be necessary to subtract like from like on both sides, until one term is found equal to one term. If by chance there are on either side or on both sides negative terms, it will be necessary to add the negative terms on both sides, until the terms on both sides are positive, and then again to subtract 15 like from like until one term only is left on each side.’ (Heath, 1910, p. 131) .

In modern terms, this passage tells us that if in a problem we are led to an equation of this type: numbers

squares

cubes

square squares... = numbers

_squares

_square squares ±...

THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING

21

we have to add or subtract the terms sharing the same /%')+ in order to reduce the problem to the case in which a +D/B%/+ equals another +D/B%/+1 that is, an equation of the type axn =bxm. To solve the problems, Diophantus often chooses some quantities involved in the statement of the problem. For instance, in the previous problem, he chooses the difference equal to 60. Then he chooses the root of one of the sought-after numbers equal to the "*%(,9)+1 which plays the role of an algebraic unknown. The other sought-after number is chosen as the "*%(,9)+ plus 3. This leads him to the equation that could be translated into modem symbols as (x + 3)² – x² = 60 where he is able 1_ to obtain the equation 6x+9=60 and finally the solution x = 8 2 . So the sought-after 1 _ 1_ square numbers are the fractional numbers 72 4 and 132 4 . Of course, these are not all the solutions of the general problem. This is why Diophantus' solutions are often seen as incomplete. However, we have to be very cautious at this point. In fact, finding out how to express (in an explicitly verbal or symbolic way) "22 the couple of members of the +D/B%/+ )7 +b4"*/+ that verify the stated condition in the previous problem (or in another problem) is a rather modem problem and not an ancient one. The problems of the _Arithmetic Books’ of Euclid's &2/9/#(+ are not concerned with the task of '/+B*%3%#$1 in an explicit form, each of the members of a certain class of numbers (e.g. the perfect numbers; see Euclid's &2/9/#(+1 Book IX, proposition 36). Another example can be the gformula’ to produce polygonal numbers found by Diophantus himself. This 'formula’ does not '/+B*%3/ the elements of a class but D*)'4B/+ as many numbers as we want (triangular, square, pentagonal numbers and so on: see Radford, 1995a). By the same token, Diophantus' problem-solving methods do not aim to find nor '/+B*%3/ all the solutions of a given problem (except, of course in the cases where the problem has only one solution) but to D*)'4B/ as many solutions as we want. The alleged incompleteness of Diophantus' solutions are relative to our modern point of view. Without taking into consideration Diophantus' own conceptualisation, Diophantus' 5*%(,9/I (%B" becomes just a mere compendium of problems solved in a way that ‘dazzles rather than delights’ and Diophantus himself appears ‘unlike a speculative thinker who seeks general ideas’ but as someone looking only for ‘correct answers’ (e.g. this is the case of Kline's perception of Diophantus: see Kline, 1972, p.143, from whom the quotations were taken). Certainly, for a Babylonian scribe, Diophantus' 5*%(,9/I (%B" would be seen as the product of a genuine speculative thinker.

THE TRACE OF PROPORTIONAL MESOPOTAMIAN THINKING IN GREEK ALGEBRAIC THINKING We are now ready to technically tackle the first question raised in the introduction of this chapter. The essence of our manoeuvre consists in showing that the functioning

L.G. RADFORD

22

of the algebraic concept of unknown in Diophantus' 5*%(,9/(%B" is too closely related to the functioning of the concept of false quantities of the Babylonian false position method to be regarded as a mere accident. On the contrary, the structural coincidence in both concepts is fully understood on the basis of the idea that the algebraic unknown was conceived as a metaphor of its correlated arithmetical concept -the false quantities. Although there are many structural coincidences between the two concepts, here we shall refer to one of them: reasoning in terms of fractional parts. Let us refer to problem number 18 from a tablet conserved at the British Museum (BM 85 196) that goes back to the ancient Babylonian period. It concerns two rings of silver. 1/7 of the first ring and 1/11 of the second weigh 1 +%B2/: The first, diminished by its 1/7, weighs just as much as the second diminished by its 1/11 (See Thureau-Dangin, 1938b, p. 46). In modern notations, the problem can be stated as 1 1 _1 follows: _ x + _ y= 1; x- x= y- _1 y. 7

11

7

11

The scribe’s reconstruction of the solution given by Thureau-Dangin (1938a, pp. 74-75) suggests that «the first ring diminished by its 1/7» is transformed into «6 times the 1/7 of the first ring ». By the same token, «the second ring diminished its 1/11»is transformed into «10 times the 1/1 1 of the second ring». The reasoning is then carried out on the above-mentioned fractions (i.e. «1/7 of the first ring» and «1/11 of the second ring»). These fractional quantities are in a 10 to 6 ratio. Therefore, by employing the false position method the scribe assumes 10 for the 1/7 of the first ring and 6 for the 1/1 1 of the second. He then adds the false assumed values and gets 16. However, he was supposed to get 1. The canonical Babylonian proportional process leads to the question of finding a 'proportional adjusting factor' which, in this problem, corresponds to the inverse of 16. The scribe finds that the inverse of 16 is 3_ 45’. To find «1/11 of the second ring», he multiplies the false value (i.e. 6) by the ‘proportional adjusting factor', 3_ 45’, and finds 22_30’. Next, to find «1/7 of the first ring» he multiplies 3_ 45’ by 10 and gets 37_ 30’. He multiplies 22_ 30’ by 11 and gets 4 o 7_ 30’ the weight of the second ring. He multiplies 37_@30’ by 7 and gets o 4 22_ 30’; the weight of the first ring. Let us now examine the Greek counterpart. In problem 6 of Book 1. Diophantus tackles the problem to divide 100 into two numbers such that 1/4 of the first exceeds 16 1/6 of the other by 20 . This problem cannot be solved by the Babylonian false position method17. However, Diophantus' method of solving the problem begins by following the Babylo nian pattern seen above: the reasoning is based on the 7*"B(%)#+ of the sought-after numbers. Diophantus takes the 1/6 of the second part as the unknown (which he calls the "*%(,9)+1 that is, the #493/* and represents it by the letter V). Thus, the second number becomes 6 times the number. ‘Therefore, he says, the quarter of the first number will be 1 number plus 20 units; thus, that the first number will be 4

THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING

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numbers plus 80 units. We want it so that the two numbers added together form 100 units. Therefore, these two numbers added together form 10 numbers plus 80 units which equal 100 units. We subtract the similar terms: 10 numbers equal to 20 units remain and the number becomes 2 units (...)’. (Ver Eecke, 1926, p. 12-13; my translation). Having found that the unknown is 2, Diophantus finds that the sought-after numbers are 88 and 12. Problem 5 of Book 1 of the 5*%(,9/(%B" shows also another example of reasoning performed on fractions of the sought-after numbers. The structural coincidence between the algebraic concept of unknown and the false quantity of arithmetical, proportional thinking can be traced to a time preceding Diophantus, as we will see in the next section.

THINKING IN TERMS OF FRACTIONAL PARTS: EARLY HISTORICAL EVIDENCE The preserved fragment of a Greco-Egyptian papyrus, dated B%*B" the first century and called Mich. 620, contains three mathematical problems with the following being one of them: ‘There are four numbers, the sum of which is 9900; let the second exceed the first by one-seventh of the first; let the third exceed the sum of the first two by 300, and let the fourth exceed the sum of the first three by 300; find the numbers’ (According to Robbins' reconstruction, 1929, p. 325). Our modern notations allow us to write the problem in question as shown in the rectangle :

The first part of the solution is not completely preserved, but it can be reconstructed from a kind of tabular arrangement or ‘matrix’ placed at the end of the solution. It is used to display the calculations and functions as an aid to help solve the problem. The ‘matrix’, which is comprised of 4 columns divided by a vertical line, suggests that the choice of the unknown, which the scribe represents as , like Diophantus did in his 5*%(,9/(%B" to designate 'the number', S"*%(,9)+T1 is 1 / 7 of the first number. It is, therefore, also the same pattern found in Babylonian mathematics.

L.G. RADFORD

24

The first sought-after quantity, which appears at the left of the first column of the table (that is, at the left of the first vertical line; see below), is equal to seven (which is an abbreviation of the whole expression «7 numbers»); the second number From that, the (found to the left of the second vertical line), is equal to eight scribe finds that the third number is 300 plus fifteen and that the fourth number is 600 plus thirty . The sum of the numbers then is 900 plus sixty which must equal 9,900. The scribe gives the answer 150, which corresponds to and then arrives at the sought-after quantities: the first one is 1,050, the second one is 1,200, the third one is 2,550 and the fourth one is 5,100.

(Table appearing in the Mich. 620 papyrus according to the reconstruction of Frank Egleston Robbins, 1929, p. 326). It is worthwhile to note, at this point, that the separation of numbers into columns, allows the scribe to divide each number into an unknown part (found to the left of the vertical line) and a known part (found to the right of the vertical line). This suggests an explicit and systematic way of dealing with the first literal symbolic algebraic language. Notice that this is basically the same pattern which is used to carry out calculations with symbolic expressions some fourteen or fifteen centuries later (e.g. Stifel, 1544). Nevertheless, we should note a difference: in the first case that of the Mich. 620 papyrus the algebraic language is seen as a ,/4*%+(%B tool (one calculates >%(, symbolic expressions); in the second case that of late mediaeval and early renaissance mathematics the algebraic language begins to be seen as a quasiautonomous object leading to a #/> (,/)*/(%B"2 )*$"#%+"(%)# (one calculates )# symbolic expressions). Many treatises will then begin to display rules on how to carry out calculations )# symbolic expressions (see the excerpt from Stifel’s 5*%(,I 9/(%B" F#(/$*"1 page 239). –

–

–

–

THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING

25

THE BABYLONIAN NAÏVE GEOMETRY

So far, our work has dealt with the #49/*%B"2 origins of algebraic ideas. We claimed that the Babylonian mathematical proportional thinking provided the conceptual basis for the emergence of elementary #49/*%B"2 algebraic thinking. There is, however, another Babylonian mathematical current which leads to another kind of 'algebra’. In fact, J. Høyrup, through an in-depth analysis of the linguistic sense of the terms occurring in Babylonian mathematical tablets, has suggested that a large part of problems was formulated and solved within a geometrical context. using what he calls a ‘cut-and-paste geometry’ or ‘naïve geometry’ (e.g. Høyrup, 1990, 1985, 1993a, 1994). In particular, this is the case of problems that have been traditionally seen as problems related to ‘second-degree equations’. We cannot discuss here, at length, the Babylonian Naïve Geometry. For our purposes, we shall just look at two examples of the new interpretation of the second-degree Babylonian algebra (see also Radford, 1996a).

Problem 1 of the tablet BM 1390 1 deals with a square whose surface and a side 18 equal 3/4. The problem is to find the side of the square .

26

L.G. RADFORD Then, the scribe cuts the width 1 into two parts and transfers the right side to the bottom of the original square.

Now, the scribe completes a big square by adding a small square whose side is ½ The total area is then ¾ (that is, the area of the first figure) plus ¼ (that is, the area of the added small square). It gives 1. The side of the big square can now be

The basic idea is that of bringing the original geometric configuration to a square-configuration. However, not all the problems can be solved by cut-and-paste methods alone. For instance, problem 3 of tablet BM 13901 deals with a square whose area less a third of its area plus a third of its side equals 20‘.

THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING

27

As Høyrup suggests, the procedure followed by the scribe is that of removing a third of the original square (fig. 1). After that, a rectangle of Width 1 is projected over the side, obtaining a configuration like fig. 2, A third of the projection is kept, which leaads to the next configuration (fig. 3). Finally, ‘in order to obtain a normalised situation (square with attached rectangle), the vertical scale is reduced with the same factor as the width of the square, i.e., with a factor 2/3 (...)’ (Høyrup, 1994, p. 13). Now the scribe can apply the procedure which solves problem 1 from Tablet 13901 discussed above. We have discussed this last example because it shows how proportional thinking also permeates the cut-and-paste geometrical thinking. Changing the scale is, in fact, a proportional idea. On the other hand, it is important to note that cut-and-paste procedures involve known and unknown quantities in a very particular way. Firstly, in Naïve Geometry, semantics plays a strong role throughout the problem-solving procedure. In Numerical Algebra, rooted in proportional thinking, the original Semantics is lost once the equation is reached (cf. Mich. 620 and Diophantus' 5*%(,9/(%B"T: Secondly, in Numerical Algebra the unknown is directly involved in the calculations. For instance, in problem 6, Book 1, mentioned above, (and translated here into modern notations in order to abbreviate our account), Diophantus performs the following calculations: 6x + 4x+80 = 100, and gets 10x+80=100. He then operates on the unknown >%(,%# a side of the equation. In other problems he performs calculations involving the unknown in 3)(, sides of the equation (e.g. Book 1, problems 7-12. In problem 7, for instance, Diophantus solves the equation 3x-300=x-20. See Ver Ecke, 1926, p. 13 ff.).

28

L.G. RADFORD

In contrast, the algebraic concepts rooted in cut-and-paste geometry (i.e. Naïve Geometry), do not seriously involve the unknown quantities in direct calculations (see Radford, 1995b, footnote 6). For instance, in problem 1, tablet 13901 seen above. the projection, and not the unknown-side, is halved. The previous discussion suggests that very different conceptualisations underlie the Algebra embedded in Naïve Geometry and the Numeric Algebra. As seen above, their 9/(,)'+ and their B)#B/D(+ are essentially different. The difference can also be seen in terms )7@ D*)32/9+: Most of the problems in Naïve Geometry deal with problems beyond the tools of first-degree or homogeneous algebra. However, it is possible to detect some interactions between both kinds of algebras. In fact, problem 27, in book 1 of Diophantus' 5*%(9/(%B"1 is a classical problem stated in the realm of Naïve Geometry. Although stated in a numerical form, Problem 27 has the traces of its old geometrical fomulation (see Høyrup, 1985. p. 103) and appears then as a numerical */B)#B/D(4"2%+"(%)# of the old cut-and-paste technique (we shall return to this point in the section, below). However, some connections between the cut-and-paste technique and first degree algebra could happen even within Babylonian mathematics themselves: this is what the solving procedure of problem 8 of tablet 1390 1 suggests. In fact, using Høyrup’s notations. we can represent the squares by Q1 and Q2 and their side by s1 and s2, respectively. The problem can then be formulated as follows:

The solution begins by taking the ,"27 of the sum of the sides. Taking a new side which is equal to the half of the sum of the sides is a recurrent idea in many of the Babylonian geometrical problem-solving procedures. The ‘half of the sum’ idea also appears in Babylonian numerical problem-solving procedures (see our discussion of tablets VAT 8389 and 8391, in Radford. 1995b). This suggests an early link between geometrical and numerical algebraic ideas.

LANGUAGE AND SYMBOLISM IN THE DEVELOPMENT OF ALGEBRAIC THINKING In this section, we would like to make a first attempt at exploring the problem of the development of early algebraic thinking with regards to language and symbolism. First of all, it is important to stress the fact that it is completely misleading to pose the problem of the development of algebraic thinking in terms of a transcultural epistemological enterprise whose goal is to develop an abstract and elaborate symbolic language. Indeed, language and symbols play an important role in the way

THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING

29

that we communicate scientific experiences. Nevertheless. their use is couched in sociocultural practices that go beyond the scope of the restricted mathematical domain. A more suitable approach to the study of the relationships between symbols and language on the one hand, and the development of algebraic thinking on the other, might thus be to analyse language and symbolism in their own historical sociocultural semiotic context. The case of Mesopotamian scribes may help us to illustrate this point. In this order of ideas, it is worthwhile to bear in mind that the oldest tablets suggest that they were first of all seen as a complementary tool to record information. The signs that formed a 'text' in the proto-literate periods known as Uruk IV and III (3300-2900 BC) reflected the key words of the messages inscribed on the clay tablet without any «syntactic relations» (Nissen, 1986, p. 329). The meaning was often suggested by the pictographic form of the sign; this is the case, for instance, of the sign SAG, 'head'. where the sign shows the profile of a head with the eye, the nose and the chin19 . The archaic pictographic writing was later replaced by cuneiform writing in all likelihood related to the needs arising from the transactions of the Sumerian administrative bureaucracy and the emergence of a new technological artefact: an oblique new stylus leaving the impression of 'nails' SB4#/%Ton the clay. The cuneiform writing was increasingly used to reproduce the oral language and when the Semitic Akkadian language became the spoken language, Akkadian was written following the cuneiform syllabic tendency. These changes brought about two important modifications: first. the way to convey the meaning of the text changed radically; it no longer relied upon pictographic insights. Second, there was a radical diminution in the number of signs. While by 3200 BC a writing system had some 30 numerical signs and some 800 non-numerical signs (Ritter, 1993. p. 14), by the first half of the second millennium Akkadian could be written with some 200 cuneiform signs reproducing the spoken language with very little ambiguity (Larsen, 1986, pp. 5-6). This point connects us with two of the most salient particularities of ancient Mesopotamian mathematical texts which have very often puzzled historians and mathematicians who attempt to understand Babylonian mathematics from the perspective of modem mathematics: firstly, the texts do not show any ‘specialised’ or 'mathematical' symbols to designate the unknowns; secondly, the texts do not display grandiose eloquence concerning the /KD2"#"(%)# of the problem-solving methods followed to reach the answer of the problem. In fact, concerning this last point, with the very moderate exception of the C/K(/+ !"(,d9"(%b4/+ '/ ;4+/1 the scribe limits him/herself20 to indicate only the calculation to be carried out to solve the 21 problem (if not, as is the case of many problems, to simply mention the answer) . The reason of what >/ see as a «silence» is not the absence of mathematical ‘specialised’ signs like 'x', 'y' (signs which are but merely inconceivable and unnecessary within the realm of Babylonian thought). This «silence» is due neither to an –

30

L.G. RADFORD

incapacity to write a mathematical expression and eventually, in the Old Babylonian period (2000-1600 BC), any algebraic one but an accepted sociocultural way to transmit and record the information which demarcates the frontiers of >,"( has to 22 be written and ,)> . To clarify this last point, remember that the tablets bearing some «algebraic» content were, in all likelihood. not produced in professional activities (e.g "BB)4#(I ing, book-keeping. surveying) but in the Scribal Tablet-Houses, that is, in the institutions created to train the future scribes. The clay tablets were, without a doubt, privileged tools in the teaching practices. However, they do not merely reflect the B)#(/#( of the teaching but they also mirror the method of teaching and its symbolic forms as well. Concerning the method of teaching, some tablets show that instruction was heavily based on the mastering of cuneiform writing; another, no less important point. was to prepare oral recitations (that were under the supervision of the second person in the hierarchy of the Tablet-House, the +/+$"2 or «elder brother». the assistant of the «father» of the School). In the written component of the scribal training, students had to copy what the teacher said or did. In many cases, the clay tablet shows a sentence (or a passage of a literary work) on one side and, on the other side, with a less confident calligraphy. a copy (visibly the student's copy) of the given sentence (see Lucas, 1979. p. 3 11 ff.) . It is not difficult to imagine the enormous difficulties that young students had to face trying to master the stylus and the rules of cuneiform writing. In a tablet. known as 'In the Prise of the Scribal Art', we can read: 'The scribal art is not (easily) learned. (but) he who has learned it need no longer be anxious about it.' (Sjöberg, 1971-72, p. 127). Another very well known text. 'Examination Text A'. that dates back to the Old Babylonian period. deals with the examination of a scribe in the courtyard of the Tablet-House. Besides the precise idea that this text provides us with an examination scene, the test uncovers some accepted teacher-student relationships such as symbolic forms emerging in the dynamic of the scribal school. The examination covers topics such as the translation from Sumerian (by that time. a dead language, as mentioned in footnote 22) to Akkadian and vice-versa. different types of calligraphy, the explanation of the specialised language (or jargon) of several professions, the resolution of mathematical problems relating to the allocation of rations and the division of fields. When the teacher starts asking questions about the techniques employed in playing musical instruments. the candidate gives up the examination. He complains that he was not sufficiently taught. Then the teacher says: –

–

‘What have you done, what good came of your sitting here? You are already a ripe man and close to being aged! Like an old ass you are not teachable any more. Like withered grain you have passed the season. How long will you play around? But. it is still not late! If you study night and day and work all the time modestly and without arrogance, if you listen to your colleagues and teachers, you still can become a scribe! Then you can share the scribal craft which is good fortune for its owner, a good angel leading you

THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING

31

a bright eye, possessed by you, and it is what the palace needs.’ (Quoted by Lucas, 1979,p.314).

In the previous passage we find mentioned explicitly the intensive work that is expected to be done by the scribe. More importantly still, we also find in the previous quotation clear instances of a symbolic form that emerges as a D"*(%B42"* social student-teacher relationship. The symbolic form is that which forces the student to show some specific attitudes: s/he is supposed to be modest and without arrogance as well as a good listener. By the same token, the same symbolic form allows the teacher to say what he said in the text. More eloquent is a passage of another text called 'Schooldays' (Kramer, 1949) in which the scribe tells us that he is caned by his teachers for doing unsatisfactory work. This symbolic form was supported by the scribes' relatives, who encouraged their sons to follow the teachers' requirements. In a tablet. the father says to his son: «Be humble and show humility before your school monitor. When you make a show of modesty, the monitor will like you.» (Quoted in Lucas, 1979. p. 321 ). We do not need to go further into the detail. It suffices to say that the aforementioned scenes clearly suggest that the teaching model relied heavily upon an incontestable imitation model concentrated on the proving of, among other things. the replication of passages from literary texts and procedural and computational mathematical skills. In this sense, the formal, semiotical content of the mathematical texts are but the mirror of the sociocultural web of relations on which schooling and. in general. all Babylonian social activities were based. To expect that the scribe product a mathematical text containing an "#"20(%B "2$/3*"%B /KD2"#"(%)# of the procedure is to expect him/her to do something that was out of all the enculturation with which s/he was provided in the Tablet-House. Explanation is, in fact, a sociocultural 23 value, not a transcultural item . Of course, Mesopotamian explanations did exist. Nevertheless, they were riot largely based upon deductive analytical principles but on metaphorical ones. A survey of literary and mythological texts is very enlightening in this respect (see for instance Kramer, 1961a, 1961b). The case of Diophantus was completely different. From the 5th century BC, arguing and explaining were two important social activities that shaped Greek thought. On the other hand, Diophantus had, at his disposal, an alphabetical language and a very well-established socially accepted system of producing and transmitting infor24 mation out of which books attained an autonomous life . Concerning algebra, even though Diophantus could use letters to represent the unknown he did not. As it is well established, the use of letters in Diophantus' 5*I %(,9/(%B" stand for an abbreviation of the word hence, contrary to our modern use, as merely an economic writing device. Even though he was not probably the first to do this. as suggested by the papyrus Mich. 620, one of Diophantus' most important semiotic contributions is to be found –

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in the peculiar use of the expression «arithmos» We previously said that he accomplished a transcendental act by including indeterminate units into the realm of calculations. But this is not to what we are referring now. Here we refer to the use of this expression of the Greek language to designate the algebraic experience that the concept of unknown carries out with itself. In the 5*%(,9/(%B" and in the specific definition of "*%(,9)+ we found this experience «uttered». While Mesopotamian scribes used the semiotic experience of everyday life and used words like the length or the width of a rectangle to handle their unknowns (using expressions like 'as much as' or 'the contribution of the length' to refer to what we now call the coefficients of a polynomial), Diophantus transformed the word «arithmos» into a more general concept. Because of its generality, this concept could apply to a great variety 25 of situations. The «arithmos» thus became a genuine algebraic symbol . Through this symbol, a numerical assimilating process of (some) geometric algebraic techniques was undertaken, leading to a new formulation of old problems and the rising of new methods in solving new versions of old problems. In turn, the new methods also allowed the Greek calculators to tackle new problems: Diophantus' 5*%(,9/(%B" contains problems which do not have any corresponding version in the algebra embedded in the Naïve Geometry. as is the case of problems concerning square-squares and cubo-square categories. Nonetheless, it is important to note that the great generalising enterprise was supported by the socially committed Greek conception of mathematics (further details in Radford, 1996b). The «arithmos»-symbolism played a particular role in a different semantisation of problems in the problem-solving phase and led to a more systematic and global treatment of problems. In order to illustrate this ideas more precisely, let us consider problems 27-30 of Book 1 of Diophantus' 5*%(,9/(%B": There is no doubt that these problems belong to an early tradition. According to Høyrup (1994, pp. 5-9), these problems, formulated, of course, in a geometrical language, go back to the surveyors' Mesopotamian mathematical tradition of the late 3rd millennium BC. These problems constitute part of a stock of problems that played a role in the rise of the Old Babylonian mathematics. Using Høyrup’s notations (modified very little), these problems can be stated as follows:

THE HISTORICAL ORIGIN OF ALGEBRAIC THINKING

33

In Mesopotamia, problem 27 was solved through the cut-and-paste technique. Diophantus does not use this technique directly. However, as was mentioned above. there are traces of the geometrical ideas in the new numerical algebraic problemsolving procedure (see Høyrup, 1985, p. 103-105; Radford, 1996a). Problem 28 appears in tablet BM 13901, problem No. 8 and in YBC 4714, problem No. 1. There is no direct Mesopotamian evidence of problem 29. (However, according to Høyrup, it is possible that originally problem 9 could be contained in a missing part of Text V of the C/K(/+ !"(,d9"(%b4/+ '/ ;4+/T: While the cut-and-paste technique docs not apply to "22 those problems in the same way, the revolutionary concept of the numerical algebraic unknown provides a similar way of tackling "22 these problems. The idea is to take the half of the sum of the sides added (or subtracted) by a certain number. (As was mentioned above, this is a procedure at the very cross-roads of geometrical and numerical ideas: see Radford, 1995b). Thus, in problems 27, 28 and 29, Diophantus represents the soughtafter numbers as «10 + 1 ‘number’» and «10 - 1 ‘number’» (in modern notations, it means 10 + x and 10 – x). In problem 30, the sought-after numbers are chosen as «X@g#493/*e@ + 2» and «1 ‘number’- 2». On the other hand, symbolism shifts the thinking from the figures themselves and makes it possible to carry out operations that do not have any corresponding sense with the initial statement of the problem. This is not the case of the cut-andpaste-technique, where it is possible to distinguish the sequence of geometrical transformations and its link with the original configuration. The introduction of the «arithmean» symbolic language provides an "4()#)9)4+ >"0 )7 (,%#=%#$ -autonomous with regards to the context of the problem. In contrast, it requires a new semantisation that has its own difficulties. This is to what Diophantus probably refers when he says, at the beginning of Book 1: ‘Perhaps the subject will appear rather difficult, inasmuch as it is not yet familiar (beginners are, as a rule, too ready to despair of success)’ (Heath, 1910, p. 129). Indeed, many of Diophantus' 5*%(,9/(%B" Scholia or comments (cf. Allard (ed.), 1983) deal with detailed explanations about the elementary symbolic treatment of the problems. Some of them use a geometric context to give a sense to the solving procedure. This is the case of a scholium of problem 26 from book 1, which explains the solution of the equation 25x2 =200X in terms of two rectangles of the same width. (Allard (ed.), 1983, p. 727).

SOME REMARKS FOR TEACHING The historical itinerary that we have followed in this work provides some information about past trends in the historical construction of very early algebraic thinking. These trends can help us to better understand the deep and different sociocultural and cognitive meanings of algebraic thinking and provide teachers

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with new paths to teach algebra in the classroom. In particular, our historical epistemological itinerary can shed some light on the didactic problem of how to introduce algebra in school. Of course, as we have pointed out in previous works, we do not claim that we 94+( follow the historical path. History cannot be normative for teaching. There are social and cultural aspects in the development of algebra that we cannot reproduce in the classroom. Furthermore, these aspects may not be necessary for our purposes. There are other aspects which could be more interesting, such as the following: (1) The epistemological meaning of algebra, i.e., that of mathematical knowledge developed around a problem-solving oriented activity, can provide some insights about the way of introducing and structuring algebra in the school; and this us to re-think, within a new perspective, the role of D*)3I 2/9+ in teaching algebra. (2) However, our study of Mesopotamian and Greek algebra clearly suggests that the specific form in which each algebra was conceived was deeply rooted in and shaped by the corresponding sociocultural settings. This point raises the question of the explicitness and the controlling of the social meanings that we inevitably convey in the classroom through our discursive practices. (3) Our epistemological analysis suggests that algebraic language emerged as a tool or technique and later evolved socio-culturally to a level in which it was considered as a mathematical object. Usually, in the modern curriculum, algebraic language appears from the beginning as a mathematical object D/* +/: Taking into account this result, it is possible to make some changes in the way of introducing algebraic language in the classroom. Following some insights of our historical studies of the development of algebra (see also Radford, 1995c), we elaborated a teaching sequence whose goal was to introduce students to algebraic methods based on different semiotic levels which culminates with the progressive introduction of symbolletters (Radford and Grenier, 1996). (4) A fourth point that we can consider, from a teaching point of view, could be the historical movement of arithmetisation of geometric algebra (which occurs in a recursive way through the history of algebra until the pre-modern epoch). Until now, the algebra embedded in cut-and-paste Naïve Geometry does not belong to the modern curriculum of mathematics; inspired by this historical research we were able to successfully develop a teaching sequence in the classroom that, through cut-and-paste geometry, has been allowing High-School students to re-discover the formula for second degree equations (Radford and Guérette, 1996).

THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING

35

( 5 ) Another aspect to consider could be the link between proportional thinking and algebraic thinking. Ratio and proportions are not presented, in modern school curricula, as being linked to algebraic thinking in the way that history suggests it happened. It seems to me that the historical metaphorical link between proportions and algebra is another interesting element to be explored in the teaching of mathematics.

Acknowledgements I would like to thank Professors Nadine Bednarz, Louis Charbonneau, Jens Høryup, Barnabas Hughes, Jacques Lefebvre, Romulo Lins, Teresa Rojano and Rosamund Sutherland for commenting on a previous version of this paper. NOTES 1

2

3 4

5

6

7

8

9

10

This research was supported by a grant from FCAR 95ER0716 (Québec) and a grant from the Research Funds of Laurentian University (Ontario). A collection of problems from Tell Harmal shows how to calculate the total price of some current items used in commercial business using silver as the ‘monetary’unit of goods. Such items included sesame, dates and lard; see Goetze, 1951, p. 153. Some such ‘non-practical’ problems will be discussed in this paper. One of them is found at the end of this section. As pointed out by Damerow, numbers in Mesopotamian mathematics are not abstract entities; they are attached to specific contexts (e.g. weight of objects, grain production). A relative detachment from the context is suggested, however, by the tables of reciprocals in the early Old Babylonian period (ca. 2000 BC) (see Damerow, 1996, particularly pp. 242-246). In what follows, we will represent the numbers in base 60; for instance a number represented by 2'' 3' 6_ 11 2 _ 2 x 60 + 3 x 60 + 5 + 2 60 60 5º 6’ 11’ means It is important to note this nuance: Thureau-Dangin was very careful with the philological aspects of the translations; in contrast, the interpretation of such translations very often had recourse to modern algebra (see Høyrup, 1996, 7-9). For instance, when discussing one of the problems of the tablet VAT 8389, Thureau-Dangin refers to the equation 40’x-30’y=8’20 and says: ‘Le scribe ne formule pas cette equation, mais i1 1’a certainement en vue’. (The scribe does not formulate this equation, however, he certainly has it in view). (Thureau-Dangin, 1938b, p. xx). As we suggested in the previous section, false quantities were generated as metaphors of true quantities. Here, a new metaphor would be used by the scribes to generate a new concept -that of algebraic unknown. We will use modern algebraic notations in some passages of our paper in order to have an idea of the problems and the methods of solution under consideration. Modern notations are not used as structural artefacts in our interpretation of ancient mathematics. Stated, of course, in a Babylonian ‘natural’ context (e.g. a stone and its weight). The Textes Mathématiques de Suse were translated by Bruins and Rutten (1961). In these Textes, there are two problems called problems A and B of text VII, related to the width and the length of a rectangle. In a recent re-interpretation made by Høyrup (1993b). Problem A of Text VII concerns the _1 equation that, translated into modern notations. reads as follows: 7 (x+4 y) - 10 = x + y , where x represents the length and y the width of a rectangle; nevertheless, our modern translation does not

36

11 12

l3

l4

15

16 17

18

19 20 21

22

23

24

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L.G. RADFORD

distinguish some of the different conceptualisations between ancient numerical operations and the modern ones. Keeping this in mind, some of the steps of the translating solution include the following calculations: 1/7[(4-1) x + y (x+y)]10 = 4 . (x+y), 3x 10+(x+y) . 10=28 . (x+y), 3x . 10 = 18 . (x+y), x . 10 = 6 . (x+y) Then, the scribe chooses x=6 and 10=x+y and he arrives at y=4. (For a complete translation see Høyrup, 1993b). One of the points to be stressed here is the fact that the calculations showed in the previous sequence are based on an (implicit) analytical procedure: the scribe's calculations comprise the unknown quantities x, y (as seen in their own mathematical conceptualisation); the unknown quantities are considered and handled as known numbers, even though their numerical values are not discovered until the end of the process. ‘J' ai mange les deux tiers du tiers de ma provende: le reste est 7. Qu’était la (quantité) originaire de ma provende?’ (Thureau-Dangin, 1938b, p . 209). The problem of the transmission of algebraic knowledge and the sources of Greek (numerical and geometrical) algebra has been studied by J. Høyrup in terms of sub-scientific mathematical traditions. (see, e.g., Høyrup, l990a). The problem of whether a conceptual organisation is scientific or not is evidently a cultural decision. In the case of the Alexandrian algebra of the 3rd century B. C., it is hardly possible to ascribe to Diophantus the whole merit ofbuilding such a theory (Klein, 1968, p. 147). Nevertheless, we can say that, in all likelihood, his contribution was conclusive to this enterprise. Freeman, 1956, fragment 4, p. 74. When reading this quotation we have to keep in mind that Heath's translation is tainted by a modern outlook. Diophantus never spoke about 'negative terms'. Diophantus spoke rather of leipsis, i.e. of deficiencies in the sense of missing objects; this is why we might remember that a leipsis does not have an existence per se but was always related to another bigger term of which it is the missing part. For a complete translation of the problem, see Heath, 1910, p. 132 or Ver Eecke, 1926, pp. 12-13. This problem can be solved by the method of two false positions. Given that this method was invented later, we will not discuss it here. Høyrup’s translation of the problem-solving procedure is the following: ‘1 the projection you put down. The half of 1 you break. 1/2 and 1/2 you make span [a rectangle, here a square], 1/4 to 3/4 you append: 1, makes 1 equilateral. 1/2 which you made span you tearout inside I : 1/2 the square line.' (Høyrup, 1986, p. 450). Although the sign could be written in a stylised format, only a few variants were allowed. See Green, 1981, p. 357. There were also female scribes, although, in all likelihood, they were not a legion! Such a scribe is the princess Ninshatapada (see Hallo, 1991 ). For an example, see the solution to the problem No. 1,tablet BM 13901, note 18. It is important to note that although Sumerian language was a dead language in the Old Babylonian period, in mathematical texts the scribes kept using some Sumerian logograms and. in repeated instances, they added phonetic Akkadian complements to some logograms as well. This rhetorical twist indeed shows a deep mastering of a very elaborated writing. Note, however, that it does not mean that the scribes did the calculations by rote. Certainly, an understanding of what they wrote was part of the task of learning (some tablets show, for instance, that a good scribe was supposed to understand what s/he wrote). It would be teleologically erroneous to think that the non-alphabetical cuneiform language of the Old Babylonian period was a delaying factor to the emergence of algebraic symbols in the Ancient Year East. The cuneiform language was a marvelous tool to crystallise the experiences, the meanings and conceptualisations of the people that spoke Sumerian and later Akkadian. Alphabetic languages correspond to new ways to see, describe and construct the word. One language is not stricto sensu better than the other: they are just different (For a critique of the alphabetical ethnocentric point of view, see Larsen 1986, pp. 7-9). In the light of this discussion, it is easy to realise that it is an anachronism to see the development of algebra in terms of Nesselmann’s three well-known stages: rhetorical, syncopated and symbolic (Nesselmann, 1842, pp. 301-306); further details in Radford, I997

ROMULO CAMPOS LINS*

THE PRODUCTION OF MEANING FOR 5N8&6M5P A PERSPECTIVE BASED ON A THEORETICAL MODEL OF SEMANTIC FIELDS

INTRODUCTION Various characterisations of algebra and of algebraic thinking have been offered by different authors (for example, Arzarello et al., in this volume; Biggs & Collis, 1982; Boero, in this volume; Lins, 1992; Mason et al., 1985). Also, many articles, books and research papers have dealt indirectly with this issue. Choices made about what algebra and algebraic thinking are have a strong impact on the development of classroom approaches and material (Lins & Gimenez, 1997), that is, the discussion of this more theoretical issue is directly related to mathematics education in the classroom. Each author makes epistemological assumptions implicitly or explicitly, the former being much more frequently the case. Almost all those sets of assumptions have at least one common feature, inherited from traditional epistemologies. The first part of this chapter deals with the analysis of that common feature, arguing that it does not allow a sufficiently fine understanding of the process of production of meaning for algebra. On the basis of this analysis, a new characterisation of knowledge is produced, leading to an epistemological model in relation to which algebra and the production of meaning for algebra are, then, characterised. The second part of this chapter consists of the examination. from the point of view of the theoretical framework developed in the first part, of two situations in which the production of meaning for algebra occurs actually or fictionally. The discussion proposed in this chapter is about ways of conceptualising cognitive activity, and the role of the little empirical evidence introduced is simply to provide a vehicle for this discussion. For the purpose of keeping a sharp focus, I will always use examples related to quite simple linear equations; I hope the reader can see in them ‘exemplary examples.’ The design model presented on page 5 1 is subject to the criticism that it —

—

* Mathematics Dept., UNESP, Rio Claro, Brazil

37 M : ;4(,/*2"#' /( "2: S/'+:T1?/*+D/B(%./+ )# ;B,))2 52$/3*"1 [X\]W: U VWWX /* 5B"'/9%B ?432%+,/*+: ?*%#(/' %# (,/ J/(,/*2"#'+:

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is artificial; I would certainly agree with this, and with the suggestion that realistic or real-life situations should be part of mathematical education in school. Nevertheless, we should not forget that many times what starts as artificial becomes quite real for pupils. even more so with the younger ones; also, many aspects of mathematical education in school are not ‘naturally’ found ‘in the streets,’ and I think algebraic thinking is one of these.

RE-THINKING EPISTEMOLOGY

h# 7"9%2%"* $*)4#' Saying what algebra ‘is’ is not a minor problem. nor one without significance within mathematics education. But we can avoid this discussion for a while, and start instead examining a much less controversial situation. I am quite sure that everyone in the mathematics education community will agree that solving an equation such as ‘3K+10=100’ is algebra. If for no other reason than because one has to deal with an ‘unknown’ expressed in literal notation, and dealing with literal expressions of this kind is algebra, even if it is not the whole of the subject. How can the task of solving ‘3x+10=100’ be fulfilled’? Certainly in a number of different ways: (1) Try different numbers, until you (hopefully) get it right.

(2) Think of 3 boxes which. together with a l0kg weight. balance a l00kg weight. (3) Think of a number, multiply it by 3, and add 10; the result is 100. Now undo it.

(4) 3 parts of a value as yet unknown, together with a part of value 10, compose a whole of value 100.

(5) Add or subtract the same number from both sides; multiply or divide by the same number. Aim at an expression of the form K =. .. . All these approaches are, in fact, so familiar that many of us tend to take them as being only different appearances of the same essence. with the likely exception of (1). I will however argue that this is riot the case. We start by characterising what the equation ‘is’ in cach case. In (1) it is a condition to be fulfilled.

THE PRODUCTION OF MEANING FOR ALGEBRA

39

In (2) it is a scale-balance. In (3) it is the do-list of a function machine. In (4) it is a whole-part relationship. In (5) it is a relationship ivolving numbers (including K) , arithmetical operations and equality. What can be done with a condition to be fulfilled? Nothing, apart from substituting numbers and checking. If the condition is changed everything changes. What can be done with a scale-balance preserving the equilibrium? Well, a lot, for instance adding the same to both sides; or removing the same from both sides, given there is enough to be removed. Secondly, and as a consequence, one can double what is on each side, but this is not necessarily as visible as the two other operations. The equilibrium is also preserved if we have 2.7 of what was on each side, but that is even less immediately visible. And with a function machine? Undo. One can, of course. use it as a condition to be fulfilled. What does not make much sense if it makes any at all is to have a result, the number on the right side. expressed in terms of the number one is seeking: this would be the case with the equation ‘4x+10=x+100.’ (Fig. 1 ) Strictly speaking, although ‘3x+10=100’ is a natural for function machine, ‘ 100=3x+10’ is not: even more disturbingly, ‘ 10+3x=100’is not natural either. —

—

A%$4*/ F

With respect to whole-part relationships, many things could be done: separate the parts (decompose the whole), for instance. Or compare that whole with some other; or make a part into a whole. Finally, as to the object in (5). one can check any textbook on school algebra. For someone who is acquainted with these five possibilities for producing meaning for ‘3x+10=100,’ it is possible to speak of metaphors and of switching from

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one to another. But we can, instcad. think of a child who is presented for the first time with ‘3K +10=100,’ and told ‘it is a scale-balance.’ It is likely that the child will learn to deal with that situation, and to use the ‘add-remove-share’ approach to solve similar equations. The question now is this: what would s/he say of ‘3K +100=10’? My guess is: ‘that one can’t be,’ as a student wrote in a similar situation, in a study we conducted (Lins, 1992). and the reason is quite simple: there cannot be a balanced scale-balance with 3 things and 100kg on one side and only 10kg on the other. If, instead, the child is told that ‘it is a function machine.’ and supposing that this is a notion already familiar for the child, a completely different situation arises. Now it is perfectly possible to produce meaning for ‘3K+100=10,’ but not for ‘4K+10=K+100,’ particularly within an activity aimed at solving this equation. Three points arise. First, the text ‘3K+10=100’ can be constituted into objects in at 1 least five different ways . Second, depending on the objects constituted, there will be a certain logic of the operations, that is. peculiar ways of handling those objects, things which can be done with them. Third. and crucially important for mathematical education, there are other equations for which it is not always possible to produce meaning in ways similar to those possible for ‘3K+10=100’. The impossibility of producing meaning for a given statement is what I call an /D%+(/9)2)$%B"2 2%9%(: This is a useful notion; as it points to the fact that producing meaning in relation to, for instance, a scale-balance. is not always a metaphorical act. A remarkable instance is the impossibility in Greek mathematics of producing meaning for incommensurability as related to numbers. For some authors the separation between geometry and arithmetic is, in Greek classical mathematics, simply a trick to avoid technical problems: however, Jacob Klein (Klein, 1968) has conclusively shown that this is not the case, and that the separation is fully consistent with the ways in which meaning for number and for geometric objects was produced in Greek classical mathematics. (see also: Lins, 1992)

2/'$/ To approach the problem of knowledge, we consider two people who have produced meaning for ‘3K +10=100,’ one of them in relation to a scale-balance, the other in relation to "2$/3*"%B (,%#=%#$i: Both would plausibly state that ‘one can take the same (10) from both sides.‘ For the first subject, it would be so because ‘if the same is removed from both sides of a balanced scale-balance the equilibrium remains.’ while for the other it would be so because ‘one can add any number (-10, for instance) to both sides of an equality, and preserve it.’ The question here is not, of course, whether or not both of them will do the same thing, but why will they do so in each case. The key problem becomes, then, is it adequate to say that both subjects

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share a knowledge? To make things more dramatic, we might want to consider a five year-old child who says that g2+3=5 because if I put two fingers together with three fingers 1 get five fingers.’ and a mathematician who also says that ‘2+3=5,’ but with a justification built from Set Theory. I think in both cases the answer is #)1 it is not adequate, but there is a further difficulty with that inquiry: we did not say what w e mean by ‘knowledge.’ Such a key question has been many times overlooked, possibly because there is a strong traditional view about the subject. or possibly because people do not see this as a relevant question within mathematical education, but most likely a combination of the two. Although many reformulations of the original notion have been produced. traditional understanding of knowledge is still bound to a classical definition: 5 D/*+)# %+ +"%' () =#)> (,"( D %7P S%T (,"( D/*+)# 3/2%/./+ %# D Y S%%T D %+ " (*4/ D*)D)+%(%)#Y "#': S%%%T (,/ D/*+)# ,"+ "**%./' "( D*)D)+%(%)# D (,*)4$, "# @ "BB/D("32/ 9/(,)'1 (,"( %+1 (,/ 3/2%/7 %+ O4+(%7%/': S;//1 7)* %#+("#B/, L"#B0 SXZZ[1 B,: VTT The above definition is usually taken as saying that to know is to have a justified true belief. Let’s examine some consequences of this definition. First, there has to be some knowledge-independent criterion of truth: some alternatives offered in this direction are objectivism, Platonism and Cartesianism. Second, and this is a very interesting aspect of the classical definition, saying that a method i s acceptable means that someone judges it acceptable; for some people casting shells is not an acceptable method for forecasting the weather, but for others it might well be. The implication i s that ‘knowing’ is a socially constructed according to it is something of an absolute situation, although ‘knowledge’ nature. With respect to the classical definition. justifications have to do with the right of a person to say s/he knows, but not with the constitution of ‘knowledge.’ The classical definition is troublesome. as shown by what is called the Gettier rightly, from the Problem, a construction in which someone would be granted technical point of view the knowing of something s/he does not know; the argument leads to the fact that the three conditions for knowing are not sufficient (Gettier, 1963). There is also the criticism, of a non-technical nature, that the classical definition rules out %9D2%B%( knowledge; I will return to this later. From the classical definition I want to emphasise the fact that knowledge, according to this definition, has the status of a proposition, being ‘that which one knows’ (for a very good and accessible discussion on the traditional view of ‘knowledge,’ see, for instance. Ayer 1986, Chapter 1); in fact, this is true also for the practitioners of the ‘implicit knowledge’ idea. Saying what knowledge %+ is not, of course, a matter of finding out the truth, but rather a matter of conceptualising things in a way which produces useful insights which to some extent agree with our general experience on the subject. We must —

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then ask: is ‘knowledge is that which one knows to be the case’ a good definition? According to that definition, we must say that the child and the mathematician share a knowledge, namely that ‘2+3=5.’ A critical instance emerges if w e consider the case of a very young child who confidently says that ‘2+3=5,’ because his father — a fine mathematician told him. The child believes what he is saying is true it is actually true, and his father is a credited authority in the field; the fact is, however, that the child is not talking about numbers not even ‘finger numbers,’ but we would be led to say that the young child shares a knowledge with his father, the mathematician. Clearly, ‘knowledge is that which one knows‘ is not an adequate notion. That the notion of ‘implicit knowledge’ relies on such an inadequate notion. can be shown by observing that someone has to say that a person has this or that ‘implicit knowledge,’ that is, at some point it assumes a propositional form. and because the person one is talking about is not aware of having the said knowledge, all we have is that proposition. If that requirement is dropped, that is, if we do not require that someone say that a person has this or that ‘implicit knowledge,’ the difficulties are even greater. as we should come to the conclusion that we all know all the things as pet unimagined by any human being. Also, we must be aware that ordinary language may be very flexible with respect to the uses of the verb ‘to know.’ Not all we know is knowledge; for instance, I might say that ‘I know John,’ but that does not mean ‘John’ is knowledge. In a similar fashion. to say that someone know-how (to do or to make) something is different from saying that a person knows-that. We can now highlight what seems to be three key aspects of knowledge. First, the person must believe in something if that is to constitute part of a knowledge s/he produces, and that implies s/he is aware of holding that belief. Second, the only way we can be sure of that awareness is if the person states it, and here I am using the term ‘state’ freely, meaning some form of communication accepted by an interlocutor: it does not have to be linguistic in form. Third, it is not sufficient to consider what the person believes and states: as different justifications with the same statement-belief correspond to different knowledge. Moreover, justifications are related to what can be done with the objects a knowledge has to do with: in the case of the child saying that ‘2+3=5,’ for instance, ‘2+3’ is the same as ‘3+2,’ once the arrangement of the fingers does not make any difference. From the point of view of a set-theory based justification, spatial arrangements are not something having to do with ‘ 2 ’ and ‘3’ or with their addition. Justifications, then, play a double role in relation to knowledge. First, they are indeed related to the granting of the right to know, and this granting is always done by an interlocutor towards whom that knowledge is being enunciated. Second, they are related to the constitution of objects. —

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Within the view I propose, =#)>2/'$/ is understood as a pair, constituted by the stated proposition which one believes to be true (the +("(/9/#(I3/2%/7T1 together with ajustification the subject has for holding that belief knowledge = ( statement-belief, justification)

A justification has to fulfil the double role indicated above: it has to be acceptable (for some interlocutor), at the same time as it constitutes, for the subject of knowledge, objects, that is, s/he can say something about those ‘things.’ Moreover, the definition establishes that there is always a subject of knowledge, rather than simply a subject of knowing, as in the traditional views; it also establishes that knowledge is produced as it is enunciated. I am not saying, of course, that knowledge is all there is to human cognitive functioning; knowledge is part of it, a substantial and accessible part. I am saying, indeed, that knowledge is characteristically human, as sign-mediated activity is.

52$/3*"

As the main purpose of this chapter is to discuss the production of meaning for algebra, my next step will be to give a characterisation for algebra; given the previous discussion. it does not seem reasonable to say that algebra is knowledge, Let’s see why. We may start noticing that we would naturally say that ‘3K+10=100 3K=90’ is algebra. But we have also seen that it is possible to produce meaning for that statement in a number of different ways; if that is to be knowledge, there is at least the justification missing. Second, it is true that, generally speaking, we identify algebra with statements D)(/#(%"220 interpretable in terms of relationships (equality, and eventually inequality) involving numbers and arithmetical operations: in order to say that ‘3K+10=100 3K=90’ is algebra. we do not make reference to which meaning is being actually produced for it. Instead of saying that algebra is knowledge, we would do better to say that it is a set of statements with the characteristics described in the preceding paragraph. Notice, however, that we know nothing about the meanings which will be produced for them by a given person, in a given situation; they may well be related to a scalebalance or to a function machine. That characterisation of algebra is operational for the purposes of research; in a later section I will show that it is also operational for the purposes of development. Algebra being a set of statements, it is (for me) text (cf. Lins, 1996). Producing 3 meaning for a statement of algebra is producing meaning for a text , and producing meaning for a text is to constitute objects from that text and relationships between

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them. Justifications. as an integral part of knowledge, play a role in the constitution of objects from the text of the statement-belief. The characterisation of algebra I propose is , then. operational in two aspects. First, in pointing out that sharing statement-belief’s in algebra is not enough evidence of sharing knowledge. Second, in pointing out that we should examine justifications if we are to identify the objects being constituted from the statements of algebra.

;/9"#(%B A%/2'+ The notions of algebra and of knowledge I have presented so far enabled us to clarify two things. First, to distinguish algebra from knowledge in which algebra contributes statement-beliefs: this is an important distinction because it allows us to account for different meanings produced for algebra. Second, it draws attention to the fact that ‘people dealing with algebra’ is a demarcation heavily marked by our own system of categories, and that we should take this into consideration when 4 examining other people’s activity of producing meaning for algebra. But there are further consequences. For instance, if Seeger’s quite interesting question (Seeger, 1991, p. 138) ‘How do teachers convert content into forms of interaction and how do students convert those forms into content,’ is reasonably reframed to ‘How do teachers convert knowledge into forms of interaction and how do students convert those forms into knowledge.’ it becomes clear that it is the dual role of justifications which play the key role. On the one hand. justifications are interactional in nature. i.e., knowledge i s always produced (,*)4$, interaction be it physically or remotely established and "%9%#$ "( interaction (Lins, 1996); on the other hand, by establishing objects they produce ‘content’ by producing meaning. The question now becomes: ‘Is there noninteractional learning?’ I am not asking, of course, whether someone can learn alone in a room; I am asking whether such lonely learning i s or is not actually interactionfree in a wider cognitive sense. When new knowledge is produced, it can be new in two ways. It can be new in that the belief stated was not a belief before its constitution into knowledge. But it also can be new in that a new justification i s produced for a statement-belief which had already been part of another knowledge, with anotherjustification. First we consider the case of someone who has produced meaning for ‘3K+10=100’ as a balanced scale, and then enunciates that (K1=) ‘I can take 10 from both sides and preserve the equality, because it i s a balanced scale.’ S/he now enunciates, for some reason (related to the person’s present activity), that (K2=) ‘I can add 90 to both sides and preserve the equality, because it is a balanced scalebalance’; that might be new knowledge, in case the person did not hold, before its enunciation, that stated belief in relation to the object he has constituted from —

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‘3K +10=100.’ Notice that the two justifications are produced in relation to the same kernel, involving a scale-balance. Alternatively, we consider that after producing K1 s/he might enunciate that (K3=) ‘I can take 10 from both sides and preserve the equality, because it is like two equal piles of stones.’ K1 and K3 have the same statement-belief, but justifications which are not only different, being in fact produced in relation to different kernels (involving a scale-balance in the first case and piles of stones in the second). In order to provide a more flexible and complete account of such differences in knowledge production, we need another construct, which I call a ;/9"#(%B A%/2': We will say that a person is )D/*"(%#$ within " $%./# ;/9"#(%B A%/2' whenever s/he is producing knowledge (meaning) in relation to a given kernel: we will refer, for instance, to someone operating within a Semantic Field of wholes and parts. Alternatively, we may say that a Semantic Field is the activity of producing knowledge in relation to a given kernel. A kernel may involve a scale-balance or piles of stones, but also wholes and parts, function machines, a straight line, areas, money, a thermometer, algebraic thinking, all sorts of fantastic creatures, colours; indeed, it may be composed by anything conceivably existing. What is known about the kernel is not ‘justifiable’ within that Semantic Field; they are 2)B"2 +(%D42"(%)#+: I am just extending Nelson Goodman’s notion of a stipulation (Goodman, 1984; Bruner, 1986). Although 2)B"2 +(%D42"(%)#+ are ‘given’ within a certain meaning producing activity, they are not necessarily basic in the strong sense proposed by Goodman for his stipulations, that is, they might well be questioned, challenged or even provided with O4+(%7%B"(%)#+ within some other ;/9"#(%B A%/2': We could perhaps say that reality is a ;/9"#(%B A%/2' with a =/*#/2 constituted by stipulations in Goodman’s sense. The notion of Semantic Field allows us a dynamic view about meaning production. On the one hand, we are able to consider how and if new knowledge comes or not to be part of a transformed kernel. On the other hand, we and if knowledge is produced in relation to a given are able to consider how kernel relate to each other. Moreover, and this is a key aspect, we are able to make full operational use of the notion of epistemological limit, already mentioned. By an epistemological limit I mean the impossibility of producing meaning for a statement within a given Semantic Field; for instance, it is impossible to produce meaning for the text ‘3x+100=10’ as a balanced scale-balance. The operational importance of this notion is to establish that: (i) every time meaning is produced there is a restriction on the horizon for further incaning production, implying that, rightly, I think as learning to produce meaning, (ii) if learning is understood teaching must also aim at an explicit discussion of the limits created in that process. epistemological limit In a later section I discuss the relevance of this construct to development and the classroom. —

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The first of the two possibilities considered a few paragraphs above (the one involving K1 and K2) I call a ./*(%B"2@ '/./2)D9/#(, the constitution of new statement-beliefs into knowledge within the same Semantic Field. The second possibility (the one involving K1 and K3 is a ,)*%H)#("2 '/./2)D9/#(1 the production of knowledge from the same statement-belief, but within a different 5 Semantic Field. A horizontal development between Semantic Fields S1 and S 2 always implies a vertical development within S2. Also, a horizontal development characterises either the establishment of a metaphor (‘3K+10=100’ (an arithmetical relationship) is as if it were a balanced scale’) or of a reversed metaphor ('a balanced scale is as if it were 6 an equation (an arithmetical relationship)'). The constructs =#)>2/'$/ and ;/9"#(%B A%/2' form the core of the C,/)*/(%B"2 !)'/2 )7 ;/9"#(%B A%/2'+ (TMSF). On this basis we can speak of producing meaning for "2$/3*" (a (/K(T as the production of knowledge from algebra within Semantic Fields. Operating within different Semantic Fields means constituting )3O/B(+ to which particular 2)$%B+ )7 )D/*"(%)# apply. New knowledge can be produced through vertical and horizontal developments. F#(/*2)B4()*+ are the source of legitimacy for knowledge, and truth is relative, but not 'absolutely relative.' From the point of view of the TMSF truth is not a notion to be applied to the statement-belief, to the proposition which we know to be the case, but to knowledge, which implies that truth is a cognitive notion, and not objectively related to 'hard facts.' To be able to decide whether or not a statement is true, certainly one must make a decision on what is being talked about; but 'what is being talked about' is constituted precisely through knowledge enunciation, and truth is, thus, relative. Justifications have a role to play in the establishment of truth, and once justifications are always produced towards interlocutors, the 'individual' cannot any longer be taken as a source of truth, as assumed, for instance, by radical constructivism. What is produced is a relativism which has cultures, through the many practices which compose them, as the domains of relative validity of any given truths. Within the TMSF the distinction between algebra and algebraic thinking becomes natural. Moreover, thinking algebraically is to be seen primarily as a consequence of cultural immersion. GLANCING AT MEANING PRODUCTION FOR ALGEBRA In this second part, I will present and discuss some empirical material and a design model for classroom activity which has been tried with pupils; all the material presented is intended only as a vehicle for discussing the notions in the TMSF, giving the reader a chance to see how the notions proposed 'work in practice.'

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;%9D2/ >)*' D*)32/9+ As part of a wider research and development project7, we have separately interviewed two pupils, presenting them with word problems. Our main objective was to investigate two things: (i) the objects with which pupils were operating; and, (ii) the role of interlocutors in the process of producing knowledge related to the solution processes. The strategy adopted was to question pupils as to the justifications they had for making the statements they did; in the particular case of the problems we will discuss, the basic ‘solution’statement always had to be a choice of operation which solved the problem proposed. Secondarily, there were other statements related to the justifications offered by them. The two pupils interviewed studied in the same state school in Rio Claro, Brazil. FEE, a girl, 13years old, and FAB, a boy 12years 7months old. The interviews lasted about one hour, during which they solved three or four problems; only one of those problems is discussed here, the Oranges&Boxes problem: (1) To calculate how many oranges will fit into each box, we divide the total number of oranges by the number of boxes, i.e., number of oranges oranges per box = number of boxes (a) If I tell you the total number of oranges, and the number of oranges in each box, how would you calculate the number of boxes used? (b) If I tell you the number of oranges in each box, and the number of boxes, how would you calculate the total number of oranges? The reason for presenting the ‘algebraic’ formula was to ascertain whether the pupils would constitute it into an object, dealing with it in the process of solving the problem; neither of them made any reference whatsoever to this formula. At first FEE seems confused about what is given and what is not. After a somewhat long exchange, she says FEE Then ...for example, there are 40 oranges, I divide byyy...(softly) Wait a minute.. .(reads the problem again, very softly). . .Yeah.. .I divi...then I would divide the.. .the total num.. .you’ll, for example, you ... for example. I have 10 boxes, 4 go into each box, then I divide the total (number) of oranges by the total of.. .how many go into each box (writes down: ‘I would divide the total of oranges, by’; looks to the interviewer) By the oranges, isn’t it?

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R.C. LlNS I What is it you want to say? FEE That I would divide the total of oranges by the oranges that go into the box. (and writes down: ‘how many oranges go into each box.’) Can I move to b ? We asked her about justifications: I . ..why is it that calculation (a division) and not another? FEE Because I am, for example, I am, I am, I have to divide among the boxes, got it, to know (‘the number’; said together with the interviewer’s next question) (. . .) FEE So, but you told me how many go into each box, OK.. .I would divide them ...for example. you say that 4 go into, there are 40, I would divide them (the ‘I would divide them’ is accompanied by a gesture: open hand, palm down, touches the table as if indicating ‘lots’), got it.. .? I Divide means what? Sharing. you’re doing? FEE Yeah, for example I put 4 in a box, 4 in the other. ‘till it’s finished.. .then I would know how many.. . (. . .) I (...)...you’re saying you thought this way, but you did a division.. .How...why did it occur to you to make a division? FEE Well, because I had for me to, for example, for me to ...to know how many.. .how many will go, for example, you have 3.. .have.. .20 oranges to put each.. .to put 5 in each box, then I'll have to divide, got it.. . ? I can’t multiply it will increase the oranges, got it? Nor add, nor subtract... I Why not? FEE Because I can’t!! (laughs) How will I add? Look, there are 38 oranges, I will add to what? (One) has to divide with the boxes, I can’t add.

A number of things emerge. FEE used specific numbers, but they were always ‘number of something,’ boxes, oranges, oranges per box, and both elements are relevant. It seems that the role played by the chosen numbers is to check a reasoning in which the logic of the operations is primarily related to sharing oranges into boxes; there is an interesting interplay between using ‘divide’ to refer to an arithmetical operation and to sharing. Moreover, the numerical division and the sharing are so closely bound that she has difficulties in explaining why she did a division when she was ‘in fact’ doing a sharing; the explanation she gives reveals that both realistic constraints (‘I can’t multiply. it will increase the oranges!’) and

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dimensional constraints imposed on the quantities involved were part of the kernel in relation to which meaning was being produced. FAB, the other interviewee, was presented with the same problem. Prompted to ‘think aloud,’ he said,

FAB I thought ... I didn’t see this bit here ...I want to know how many oranges go into, then I.. .how many oranges would take.. .and would divide ...I would take all the oranges and divide ...by the number of boxes.. .(frowns) No.. .no...I wouldn’t have the quantity of boxes.. . I OK, then.. .but you want to know the quantity of boxes.. .what do you know’? You know how many oranges go into each box, how in any oranges altogether, you want to know how many boxes you need.. . FAB ...(lo oking at the problem) I would take the whole and put . . .I would divide by the number of bo.. .of oranges that go into.. . FAB (reads what he had written down) ‘I would take the quantity of oranges and divide by the quantity of oranges that go into each box.’ I Are you sure that you would get the number of boxes.. . FAB (nods, and moves to ( 1 b)) I What did you think that took you to this conclusion? FAB (smiles) The.. .(smiles). . .I.. . For a while he did not say anything, so we asked him to play as if he had to explain to a cousin why he did the problem that way; FAB said it couldn’t be because his cousin was too young to go to school. We decided to adopt 'a friend' instead of the cousin. Imagine your friend is there, at your side. and he asks ‘Listen, FAB, how do you know it? How did you think to.. .solve the problem?’ FAB . ...Well, I thought if I had the oranges with the box. I Hmm. Try to show me.. .How did you imagine it? Try.. .if you want to make a drawing, anything with the hands, to speak. whatever you want. FAB (drawing round shapes on the paper) I imagined I had a pile of oranges. I Hmm. FAB Then ...I took a box (draws a square 'u')...which would hold ...a certain amount (draws some round shapes inside the 'box') Then I thought ‘If I divide this amount of oranges (points to the shapes outside the box) by the amount which is inside (points to the box). . .which goes into here, 1’11find out.’ I

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FAB’s solution is substantially different from FEE’S in at least one key aspect: he #/./* mentions any specific numbers. The objects he is operating with are related to boxes and oranges. More precisely, he seems to constitute the following objects: unit oranges, a pile of oranges, and a box. Those objects have properties; for instance, the pile of oranges can be counted or separated into boxes. The logic of the operations engendered by his construction did not depend on specific numbers. It seems he thought of the operation ‘separate the oranges in the pile into smaller, equal, groups,' and only then indicated the arithmetical operation division which, as a tool, would be used in order to do an actual calculation. The use of an arithmetical operation is subordinated to the logic of the operations proper to the objects constituted. The boxes-oranges kernel is so solid that when first approaching the problem he says ‘I would take the whole and put...,’ and immediately changes to ‘I would divide by (. . .)’ (my emphasis); he is thinking around a kernel of boxes and oranges, but the problem says 'how would you B"2B42"(/:’ Another aspect of interest is that, although he had already hinted that he was operating with 'boxes and oranges' objects, it took a long exchange before FAB felt he could give the justification he apparently had in mind, and it is remarkable that this exchange involved precisely a proposed change in interlocutor, from the interviewer to a peer. The slip-of-the-tongue 'I put,' provides a strong indication that the justifications given later were not some 'rational reconstruction.' but rather a true enunciation, to a new interlocutor, of a faithful account of the 'actual' solving process. FAB also solved (1b) correctly, but this time it took him no time to produce a justification within a Semantic Field of oranges and boxes: I The same thing: how would you explain it to your friend? FAB I had ...as I imagined, that I had the boxes, some ten boxes. Then I would do the opposite to that (points to (la)). I would take as if I was going to count, but instead of counting I would find out how many oranges in each box and would do times. By the number of boxes used. ... I Then you did a multiplication...Why? FAB Because it's quicker, isn't it, than counting one by one. I But it is the same thing you're doing. just that to calculate... FAB Yeah.

A possibility for FAB’s need to specify 'some ten boxes,' is that in the ordinary experience of most people with boxes there are never too many and these can be precisely quantified without difficulty, while with respect to piles of things one is

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almost never expected to have a precise or near-precise idea of the quantity; put in other words, D%2/+ )7 )*"#$/+ "#' +/(+ )7 ,)+/+ 9"0 ,"./ ,"' '%77/*/#( D*)D/*(%/+ >%(, */+D/B( () b4"#(%7%B"(%)#: In solving (la) it is just natural that the number of boxes is not determined at first, and the other objects (a pile of oranges and lots of oranges to be put into boxes) do not suggest the need of a perceptual at-a-glance quantification. FAB clearly constituted a distinction between the operation actually carried out (counting of a number of equal lots) and the arithmetical operation used as a tool to evaluate the result of the counting: FEE, however, did not. Altogether, the interviews showed us different processes of meaning production for a text involving oranges and boxes. Numbers were constituted as different objects in each case, that is, they had different properties and played different roles. Distinct logics of operations were in place, but in neither of the two cases properties of the arithmetical operations played a part in these logics of operations, that is, arithmetical operations were not made into objects.

5 '/+%$# 7)* /KD2%B%( O4+(%7%B"(%)#+ In this section I will argue that justification, as a constitutive part of knowledge, has to become an explicit part of classroom environments. Presenting pupils with 'problems to solve' will focus the activity on producing a solution, and it is only natural that trying to get them to discuss their methods starting from a solutiondriven problem requires some considerable effort on the part of the teacher. Classroom common-sense, built both from tradition and from some scientific conmion-sense, suggests that the natural direction is from 'concrete' to 'abstract'; one possible interpretation here is that 'concrete' may refer, for instance, to problems with specific numbers, while 'abstract' would refer to problems with generic numbers. Freudenthal has already argued against such conception, pointing out that it is not true that generality is always achieved through generalisation (Freudenthal, 1974). More particularly, this comment was prompted by an analysis of Soviet work on the early introduction of generic, literal, expressions to pupils, and in the cases he examined, 'early' meant the first grades of primary school. If we take in particular the pioneer work of V.V. Davydov (e.g., Davydov, 1962), the most striking feature is a conceptual shift through which he departs from the accepted notion that 'numerical literals' (our terms) can only be made meaningful as 'variables', as generalisations of specific numbers. What Davydov proposes is to work from generic quantitative relationships, as one would find in a situation involving cars and trucks in a parking lot: 'In a parking lot there are two kinds of vehicles: some are trucks and some are cars. If all the cars leave, which vehicles are left?'

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A%$4*/ V As we have already indicated in Lins (1994), Davydov’s work is based on the notion that even if the original 'support' is provided by a trucks and cars situation, what one is in fact dealing with is the 'true' nature of simple algebraic relationships. i.e., pupils are given the chance to work with an embodied version of the essential whole-part relationship. From the point of view of the TMSF, however, what Davydov is proposing is that pupils produce meaning for literal expressions within a Semantic Field of cars and trucks, and then use these expressions as a departure point for beginning the development of Algebraic Thinking, as meaning for new statements gradually conies to be produced in relation to statements already made meaningful, rather than in relation to the original situation (kernel). This shift is not treated explicitly in Davydov’s activities, but it could have been. Based on Davydov’s original idea, I have developed an activity in which literal expressions are made meaningful within a given Semantic Field (of water tanks), and then a deliberate shift in the way meaning is produced for new expressions generated is proposed to pupils. This activity has been tried with sixth-graders in Brazil, and an overview of the results is presented in Lins (1994), will not be discussing here actual pupils’ work, but the overall shape of the approach proposed. The activity is introduced with the following text. What is being proposed is that the tanks situation will constitute a kernel in relation to which meaning will be produced for various statements. Objects or could be buckets and tanks. Implicitly, there is constituted in that kernel are also water, or some other liquid; another local stipulation, suggested by the drawing, is that the two tanks are of equal size. One possible first statement is. S1 ‘The tank on the right has more water than the tank of the left.’ There are, however, at least two different justifications for the enunciation of this statement-belief: ‘The line of water is higher on the tank on the right,’ or, J1A —

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‘9 buckets are needed to fill up the tank on the left, but only 5 are

needed on the right.’ Within the TMSF, the knowledge K1A= (S 1, J1A is different from K1B= (S1, J1B). The difference is not just a formal one; in K the drawing itself becomes an 1A object which has a place in cognition, while in K1Bthere is only reference to the 'verbal' part of the text. Notice that we are not claiming that the subject enunciating K1 B has never or will never constitute the drawing into an object: it is just that in the enunciation of K1 B that object does not seem to exist. While K1 A seems to have a more qualitative nature, K1 B seems to have a more quantitative one. To appreciate more fully the characteristic difference between and K1 B, we may consider whether similar justifications could be used in relation to the following question: ‘If I add one bucket of water to the tank on the left. will there still be less water in it than on the tank on the right?’ Operating with a quantitative relationship there would be no difficulty in providing an affirmative answer. ‘If one adds a bucket of water to the tank on the left, it will still have S2 less water than the tank on the right.’ ‘8 buckets will still be needed on the left, but only a on the right’ J2 Operating only with an object constituted from the drawing, however, it is impossible to produce an answer. It is possible, of course, to consider that the top 'white' space on the right corresponds to 5 buckets, and to estimate the 'slice' corresponding to a bucket, using that estimate to conclude visually that there would still be more water on the right than on the left. That operation, however, depends on also constituting 'a number of buckets' as an object. Returning to the activity, I would propose that the pupils produced valid statements together with justifications. It became more comfortable to assign singleletter names to the objects being referred to; thus. 'buckets' became 'b,' and equality was naturally indicated by '='. As to the tanks, we finally agreed on 'T'. The amount of water in the tank on the left was named 'X', and that in the tank on the right, named 'Y'. All these choices were made together with the pupils, and they seemed to have added no further difficulty to the activity. The pupils in question were Brazilian sixth-graders (12-13 years old), who would have had by then an introduction to simple equations using x’s or y’s. One might expect to get expressions like 'X+9b=Y+5b,' which are not to be understood as 'equations.' At first it is natural to get justifications which all refer back to the kernel, for instance: S3 ‘X+9b=Y+5b’ ‘If 9 buckets are added to X, we will get a full tank, and the same J3 happens if we add 5 buckets to Y.’ Or,

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S4 J4A

‘X+4b=Y’ ‘If 4 buckets are added to X, there will be 5 buckets missing on the left. and this is what is missing on the right.’

The enunciation of the knowledge (!4,"4A) produces meaning for S 4 within a Semantic Field of tanks, Meaning for a statement is produced by the constitution of objects froin that text. One could also produce the following justification for S4: J4B ‘If 9 buckets are missing on the left, and 5 buckets missing on the right, this means that Y has 4 buckets more than X.’ which is, of course, different froin J4A. The reason for eliciting justifications which refer back to the kernel is that the statements must first be made to correspond to objects already constituted, and the kernel, with its objects constituted through local stipulations plays the psychological role of reality: that is, of course, an approach radically different from objectivist theories of meaning, for which ‘hard core reality’ objects arc the things in which meaning is 'anchored'.8 Once meaning is established for a set of statements. within a Semantic Field of talks, it is possible to suggest that pupils explore another way of producing correct statements about the tanks situation, and this is done by examining possible relationships between already established statements: how could one ‘reach’ the already meaningful starting from the already statement ‘X+4b=Y’ statement ‘X+9b=Y+5b’? Our algebra-educated minds would meaningful certainly say, quite naturally, ‘Take 5b from each side.’ Before that can be taken as a natural step, however, we must consider that the very task proposed involves two crucial steps: (i) that the statements themselves become objects; and, (ii) that pupils’ thinking shifts quite strongly froin the tanks situation. Let’s examine the consequences of that. First, in order to constitute the statements into objects. we must be able to say something about the properties they have "+ >,)2/ +("(/9/#(+Y it is not enough to say what their constitutive elements "*/ nor what the statement says about those objects. But pupils are precisely being required to say something about what does not exist yet for them. +("(/9/#(+ 4+ )3O/B(+: There seems to be an epistemological paradox here. Second, any direct transformation froin ‘X+9b=Y+5b’ to ‘X+4b=Y’ will produce a new meaning for the latter. But meaning has already been produced for ‘X+4b=Y’; why would a pupil take aboard this new meaning instead of, or even in addition to, the original one, naturally produced by taking ‘X+4b=Y’ directly in relation to the kernel? There seems to be a didactical paradox here. —

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Before we set out to solve those paradoxes, a key question must be answered: why would >/ want to introduce a new way of producing meaning, when the previous one seems so promising? The answer to this question is as simple as it is crucial in solving the paradoxes: we do this because >/ >"#( pupils to be able to operate both ways: >/ >"#( them to be able to produce meaning linking a statement to a kernel and >/ >"#( them to be able to produce meaning by performing direct transformations of statements. But this is essentially a decision made on a cultural basis: that is what our culture expects from someone who is performing the function we are; there is no plausible reason to believe " D*%)*% that in any other given culture people are expected to be able to produce meaning through the direct transformation of statements. What makes this a key assumption in the solution of the paradoxes, is precisely that our function as interlocutors will provide the reference for the intention to constitute whole statements into objects, at the same time we, as interlocutors, are the agents of trying to get the pupils to engage in the activity of producing meaning in the new way we are proposing. The paradoxes are solved, then, by observing that there is an intermediate step in which the %#(/#(%)# to engage in a new activity is the key factor, and during which however brief it is authority plays a crucial role. It is never too much to observe that I am using the notion of authority just as to indicate a reliable point of reference. The statements are, then, first constituted into ‘objects whose properties I do not know (although I know they exist because my reliable interlocutor indicates so).’ This is not very different from pupils listening to a lesson about a distant country: ‘There is a place where people.. .’, and that is precisely what constitutes the country and the people the pupils engage in thinking about. It is certainly essential that they have already produced some meaning for 'people' and 'country.' Both epistemological and didactical paradoxes are solved at once: the first meaning for a statement as a whole is precisely 'statements can be treated as a whole', and the justification is the teacher’s authority, although probably nothing else is at first known about >,"( really can be made with them; on the other hand, pupils engage in that activity %7 (,/0 ') -because the teacher represents if s/he does the legitimacy of the newly proposed way of producing meaning, and because pupils want to belong to a social practice in which that way of producing meaning is legitimate and desireable. The paradoxes were rooted, in fact, in conceiving the possibility of a transition from the old to the new. But the formulation I present makes clear that it is not the case of a transition, but actually it is the case of a rupture, and that the rupture is promoted within a process of interlocution: ‘let’s do it differently,’ and someone has to have a reason for doing it differently.

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The next step would be the production of new correct statements in relation to the tanks situation, but with the requirement that two justifications are produced: one in relation to the kernel and another produced by a direct transformation of a previously accepted statement. The reason for this is precisely to make explicit the existence of two different ways of producing meaning for new statements. It is possible to produce meaning for the statement, S5 ‘X-1b=Y-5b’ both with J5A ‘If I take 1 bucket from X, 10 buckets will be missing; the same happens if I take 5 buckets from Y, because 5 buckets were already missing, and I am taking another 5.’ and with J5B ‘Take 5b from each side of X+4b=Y, which I had already established as correct.’ As soon as the property ‘one can take the same from both sides of a statement’ is established, it is possible to produce statements like ‘X-40b=Y-44b,’ which although correct according to the properties of the objects ‘statements,’ cannot be made meaningful within a Semantic Field of tanks, simply because it does not seem plausible that one can remove 40 buckets from X. It is now possible to produce a strong distinction between the two modes of producing meaning, on the basis of the fact that there are objects in one case which cannot be made into objects in the other. The crucial step is, I insist, to produce a rupture, not a transition.9 There are several possibilities to follow from here. One of them is to start working on producing ‘target-statements’ from statements already produced, for instance, from ’X - 1b=Y-5b’to produce a statement of the form ‘X=...’ or ‘Y=...’ or, later, of the form ‘b= ...’. In one of the groups, two solutions appeared to the ‘targetstatement’ ‘b=...’ from ‘X+4b=Y’: ‘b=Y-X-3b’ and ‘b = j \ k ' The latter came 4 from a student whose mother is a mathematician, who gave her ‘a hint.’ The activity described above is intended to provide a design model. On the basis of the rationale for that design is the fact that meaning can be produced for algebra in a number of different ways. It is meant to indicate that the so-called ‘concrete embodiments,’ if taken together with the assumption of the possibility of a transition the traditional didactical effort to produce a ‘silent transition’ is likely to hide from pupils the fact that they are being required to produce meaning within a new, and distinct, Semantic Field. Moreover, it indicates how it is possible to overcome such problems and still keep the possibility of starting from already constituted, familiar, kernels. Three axis were taken into consideration on the design proposed: (i) one which goes from +)24(%)#I'*%./# to 9/(,)'I'*%./# activities, the latter being characteristic

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of the Tanks activity; (ii) language, representations and notations; and, (iii) the logics of the operations, that is, kernels and Semantic Fields. SUMMARY AND CONCLUSION One key thing has been shown in Parts 1 and 2: that the characterisation of knowledge adopted by traditional epistemologies is inadequate, in particular for mathematics education. As an alternative, I have presented a characterisation of knowledge which incorporates as a constitutive element the justification a person has for believing that something is the case. The notion of knowledge as a pair (statement-belief, justification) is the basis for the construction of the Theoretical Model of Semantic Fields. Within that model, the production of meaning is an activity which happens around kernels constituted by local stipulations. That activity constitutes Semantic Fields. Justifications have the double role of constituting objects and of taking part in the process of a person being granted with the right of knowing that such and such is the case. Objects are, then, constituted within Semantic Fields. Whenever objects are constituted, there is a particular logic of operations which applies to them, i.e. what can be done with them. Knowledge is always enunciated to an interlocutor. Within the Theoretical Model of Semantic Fields, interlocutors are an essential part of cognition, as the production of meaning is always directed towards an interlocutor. When we produce meaning we are speaking to an interlocutor, either internal or external. Characterising algebra as a text, rather than as knowledge, allowed us to account positively for different meanings produced for it, without having to slide into a hierarchy in which 'official' ways of producing meaning arc at the top. There are two key consequences: (i) we are not forced any longer to treat children’s cultures nor any other culture as 'lacking';and, (ii) we are able to characterise the process by which meaning production might if not properly dealt with constitute limits for pupils’ learning. With respect to item (i), it is important to point out that any epistemology which characterises what is on the basis of the very culture within which it has been produced, is clearly unable to be of much use in helping us to move forward, to go beyond limits historically and materially produced. A number of educational consequences can be drawn. First, that instead of simply looking for ‘meaningful learning’, we must take into account the possibility of different meanings, and that we may be particularly interested in getting pupils to produce meaning in a specific way; in the case of algebra, the various ways of producing meaning are of interest, but we may be particularly aiming at getting them to think algebraically, although not to the exclusion of other possibilities. —

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Second, it is possible to organise classroom activity around different modes of thinking, rather than in terms of 'content' as given in mathematics itself. One key aspect of this new organisation, is that contents would be #"(4*"220 integrated. For instance, around 'thinking with wholes and parts' one would have fractions, some algebra, some geometry. Around ‘thinking with scale-balances’ one would have some physics, some algebra, some measurement. I am not mentioning the possibility that contents from outside mathematics are also taken aboard. In a sense, organising classroom activity around modes of thinking — as I characterise them — has some flavour of project-oriented approaches; I believe the two ideas can and should be thought of together. Third, that by getting pupils to make justifications explicit we may go far beyond the simple possibility of checking whether they ‘really’ know what they are saying. Through this process pupils and teacher will be able to produced +,"*/' meaning and +,"*/' knowledge. In terms of the teaching process, that enables the teacher to identify and approach, from inside, situations where learning is not occuring. Whenever the teacher has to deal with it only from the outside, there are two possibilities: (i) insist on the approaches already used, as if pupils needed a second chance to ‘see’ what they did not in the first try; or, (ii) leave it to the pupil, perhaps in the sense of assuming that the pupils was not yet ready to learn those ideas being proposed. In both cases the teacher is very much passive. But by entering into the pupils’ world of meanings, and by making explicit that at some points new ways of producing meaning are being proposed, both teacher and pupils become truly active in the constitution of a common, shared discourse. The sharing of statement-beliefs "#' justifications, on the other hand, are not seen any longer only as a politically correct attitude, in the solidarity sense of sharing; that is certainly important, but there is now also the fact that such a process is an essential, constitutive part of learning, as it is through this that the legitimacy of given modes of thinking is eventually established for those ‘listening’. I had already mentioned the role of the teacher as an interlocutor; the role played by pupils among themselves is quite similar to that of the teacher. Fourth, and last, there should be a shift away from the usual ‘concrete to abstract’ notion. The suggestion is that we stop thinking of scale-balances and function machines, for instance, only as intermediate steps in the road towards ‘what algebra really is.’ Instead of asking the question ‘how to bridge the gap,’ perhaps we would rather acknowledge that there is no possible ‘bridge,’ there is no transition. The idea of a transition is certainly rooted in the notion that there is a higher level of thinking to be reached from lower ones. Davydov’s work, and my own, shows that changing the perspective with respect to ‘concrete to abstract’ allows us to produce powerful classroom approaches.

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Although widely aimed as a model for epistemology, I think that the Theoretical Model of Semantic Fields provides a simple, yet powerful, tool for research and development in mathematics education, as well as for guiding classroom practices and for enabling teachers to produce a sufficiently fine, thus useful, reading of the process of meaning production in the classroom. Finally, I would like to emphasise that the Theoretical Model of the Semantic Fields is not a 'local theory' aimed only at the production of meaning for algebra; it is similarly applicable to all parts of Mathematics. In fact, it applies to any process where the production of meaning occurs, but this is certainly not the place for such a discussion.

Acknowledgement I would like to thank Alan Bell, Paolo Boero, Ole Skovsmose and Rosamund Sutherland for their insightful and sharp comments on many of the ideas presented here. I am also indebted to the other members of the PME Algebraic Processes and Structure Working Group.

577%2%"(%)# -#%./*+%(0 )7 ;") ?"42)1 6*"H%2 NOTES 1

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It is sufficient to say, at this point, that an object for a person is anything which that person can say something about. Strictly speaking, ‘3x+10=100’ might be constituted into objects even before any ‘mathematical’ meaning is produced. once the person may say that ‘3x+1 10=100’ can, for instance, be typewritten or handwritten, and in large, medium or small size. This is, however, a subtle aspect which will not be discussed any further here. I n Lins (1 992), I have characterised Algebraic Thinking as thinking arithmetically, internally, and analytically. Thinking arithmetically can be understood as ‘thinking in numbers'; thinking internally means not modelling back those numbers as some other objects (eg, measures or wooden sticks, currency or areas), ie. characterising them only as objects having given properties in relation to the operations and to equality and eventually inequality; and. thinking analytically means treating unknown numbers exactly as if they were known. Put together, these three conditions point to objects. numbers. which are known only as objects we operate on with the arithmetical operations. As suggested before, by a text, from here on, I will mean not only written text, but any residue o f an enunciation, sounds (residues of utterances), drawings and diagrams, gestures and all sorts of body signs. What makes a text what it is, is the reader’s belief it is indeed a residue o f an enunciation, that is, a text is framed by the reader: also, it is always framed as such in the context of a demand that meaning be produced for it. For instance: it may seem natural for us to place equations and functions close to each other, but this possibility depends on constituting them as objects with certain common features; such a constitution may not be, however, within the horizon of a given person’s ways of producing meaning for those objects. In the work ofthe Dutch group of Utrecht (see, for instance, van Recuwijk, 1995), we find the notions of vertical and horizontal mathematisation. Although similar, I would like to point out that vertical and horizontal devlopments within the TMSF are much more general notions than their Dutch counterparts, particularly as they are not aimed only at mathematical meaning In particular, the Dutch

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R.C. LlNS version characterises ‘mathematisation’ in terms of notation, but without clarifying whether or not this carries with it-within their model-‘meaning.’ This example is given assuming that meaning had first been produced within a Semantic Field of a scale-balance. A metaphor establishes the first Semantic Field as ontologically more primitive, while the reversed metaphor produces a restructuring of the ontological building: ‘now I know that in fact ... ’ Parts of this section have already been reported at PME XIX, Recife (Brasil), 1995. I am particularly indebted to Geraldo Garcia Duarte Jr., Rosamund Sutherland and Luciano Meira, for their insightful comments on the interviews discussed here. Producing meaning for algebra: a research and development project in teaching and learning, a cooperation project conducted under the direction of Rosamund Sutherland (University of Bristol, UK) and the author (Dept. of Mathematics, UNESP-Rio Claro. Brazil); the project is partially funded by CNPq (Brazil), grant 530230193-3, and by the British Council. An illuminating instance of the need of familiar kernels is found in the use of ‘examples.’ For instance, when I teach Group Theory, it is common that when presented with the definition of a Group my students are completely unable to say a word. After discussing a few examples, however, many of them become able to produce statements justified within a Semantic Field of the formal definition. Familiarity with the examples, here, allows them to say things. On a more technical note, it is worth indicating a general mechanism of production of new Semantic Fields, namely, the introduction of a new operation on objects previously constituted. In the history of mathematics we find a prime example of that mechanism in action, when Wessel introduces a multiplication of directed lines, objects which are first constituted as displacements (Wessel, 1959), and, as a consequence, produces new objects, which are distinct from the previous ones; in this particular case, Wessel then associates these new objects with complex numbers, by showing that the addition and multiplication of directed lines have the same properties as those of complex numbers. It is interesting that he is not seeking a foundational model for complex numbers; instead, he is trying to develop a ‘geometric calculus’-after all. he was a surveyor-, and what he shows is that complex numbers are helpful in dealing analytically with directions, that being his main objective.

FERDINANDO ARZARELLO¹, LUCIANA BAZZINI ² AND GIAMPAOLO CHIAPPINI ³

A MODEL FOR ANALYSING ALGEBRAIC PROCESSES OF THINKING

INTRODUCTION It is very well known that students show difficulties in learning the symbolic language of algebra; many authors have pointed out that a major problem consists of students’ incapability of relating symbolic expressions to their meaning: the roots of many misunderstandings lie in the inadequacy of such a relationship, leading to incorrect performance and blind manipulations with algebraic symbolism (see Sfard (1991), Kieran (1992)). Sometimes students do not only ignore the correct meaning of formulas and concepts, but even invent fresh meanings which surrogate the authentic ones. From a didactic point of view, it is very hard to convince students that they are wrong, in so far as the invented meaning often has its own justification, generally rooted in previously learned models, perhaps working appropriately in their own context. Hence, it may happen that the teacher and the student do use the same words, but with different meanings: a genuine comedy of errors is thus generated. A consequence of this is that many secondary school students do not master the sense of those symbols, which they have learnt to handle formally (for example, see the USA National Assessment of Educational Progress Report in Brown et al.( 1988), also quoted by Kieran (1994)). On the other side, some students, even if clever ‘algebraic calculators’, seem not to be able to see and use algebra as a means suitable to understand generalisations, to grasp structural connections and to argue in mathematics (see Laborde (1982)). Existing literature has shown the possibility of taking instant pictures of students’ difficulties, but it is not so easy to find the way to analyse the cognitive processes involved for longer periods of time and from a more global point of view and, consequently, to provide suitable suggestions for teaching. 1

Dept. of Mathematics, University of Turin, Italy.

2

Dept. of Mathematics, University of Pavi, Italy. 3 Istituto per la Matematica Applicata, C N R , Genoa, Italy 61

M : ;4(,/*2"#' /( "2: S/'+:T1?/*+D/B(%./+ )# ;B,))2 52$/3*"1 ]X\lX: U VWWX /* 5B"'/9%B ?432%+,/*+: ?*%#(/' %# (,/ J/(,/*2"#'+:

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Algebraic thinking is recognised as being inseparable from the formalised language by means of which it expresses itself. However, it is reductive to believe that algebraic thinking lives only at this level; in this case everything would be reduced to a manipulative mechanism, which often does not work in the hands of students. In accordance with many people, particularly with Vygotsky’s theory about the nature of links between thought and language, we consider algebraic thought and language as two intertwined and mutually dependent aspects of the same process. In fact, analysing the word’s central role, Vygotsky states that it would be incorrect to consider thought and language as separate or independent, in spite of their separate origin. In particular, he stresses that a word’s meaning is a linguistic and intellectual phenomenon which evolves over time. This statement is of special interest when applied to the meaning of algebraic expressions; language is conceived of as a thinking tool progressing in time: in turn, the evolution of thinking inspires the use of more complex and sophisticated linguistic forms. It is our concern to expose here a theoretical model which is suitable for analysing algebraic thinking from such a point of view. This model has been developed over the last years by the authors in a n ongoing joint research project (see Arzarello et al. (1992), (1994)), and has as an empirical basis the behaviour of hundreds of students, from the 8th grade up to University, observed while solving (pre-algebraic and algebraic) problems. The chapter collects and exposes in a more systematic way the contributions given by the authors at PME Conferences, both in presentations and in the m)*=%#$ 8*)4D )# 52$/3*"%B ?*)B/++/+ "#' ;(*4B(4*/+: The emphasis is on the theoretical analysis, while didactic consequences, which arc the present object of investigations by the authors, will be discussed elsewhere.

;/#+/ "#' '/#)("(%)# )7 "# "2$/3*"%B 7)*942"P (,/ (*%"#$2/ )7 A*/$/ It is our aim to present a precise theoretic analysis of the meaning of symbolic expressions in algebra as a concrete tool to describe the dynamics of typical algebraic processes and misconceptions. Our starting point is the ideas of Frege on semantics (see Frege (1892a,b; 1918): in fact they seem suitable for looking at the interpretation of symbolic expressions of algebra, insofar as learning is concerned. In particular, we shall distinguish between Sinn (sense) and Bedeutung (reference, denotation, also meaning, but the English translations are ambiguous) of an expression (Zeichen): the Bedeutung of an expression is the object (Gegenstand) to which the expression refers, while the Sinn is the way in which the object is given to the mind (Figure 1), or in other words, it is the thought (Gedanke) expressed by the expression. Everyone knows the example of Frege concerning the two different senses of Venus, namely as Esperus, i.e. the

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Frege’s Semiotic t riangle A%$4*/ F

night’s star, and as Phosphorus, i.e. the morning’s star: the two expressions have the same denotation. that is the planet Venus, but different senses. Also in Mathematics there are expressions whose senses are different but which have the same denotation. For example, the expressions 4x + 2 and 2(2x+ 1 ) mean a different rule (sense) but denote the same function; likewise, the two equations (to be solved in R) (x+5)² = x and x²+x+1=0 denote the same object but have a different sense. So mathematics, as natural language, is full of expressions which have the same denotation but incorporate different senses. The most ‘evident’ sense of an algebraic expression represents concisely the very way by which the denoted object is obtained by means of the computational rules expressed in the formula itself; we call it the "2$/3*"%B +/#+/: For example, the formula n (n+ 1) in the universe of natural numbers expresses a computational rule, by which one gets the (denoted) set A = {0.,V1 6, 12, 20, ...}. But the same formula is able to incorporate additional senses. apart from the algebraic one. In fact, it can be used in different knowledge domains, mathematical or not, each generating (at least) a new sense, depending on the nature of the domain. For example, the expression n (n+1) in elementary number theory has the sense of ‘product of two consecutive numbers’. whilst in elementary geometry it may stand for the area of a rectangle of (integer) sides n, (n+1). We call the B)#(/K(4"2%+/' +/#+/ )7 "# /KD*/++%)# a sense which depends on the knowledge domain in which it lives (as such, it is different from the algebraic sense): in fact the above formulas express different thoughts, with respect to the different contexts where they are used (see the discussion in Frege ( 1918)).

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The variety of senses that an expression can have is shown in a very direct way by the formula itself, because of the %'/)$*"D,%B features of the algebraic language: which means that the formula mimics in its own form and shape the main relationships among the different objects involved. In the above example, the fact that the elements of A are obtained from the product of a number and its successor is shaped ideographically in the formula itself (compare it with the English sentence which expresses the same fact. where the representation i s purely symbolic, without any ideography). Ideography allows for changes of sense by suitable manipulations on the shape of the formula (for example, n (n+1) ---> n²+n). The power of algebra consists in the multiple senses which are incorporated by the same formula and/or which can be obtained by syntactic manipulations on it; whilst its didactic drama resides in the complete imbalance between senses. denotations and expressions, which make the status of algebraic signs very obscure for students. Let us make some examples. First, senses may change without a corresponding change either in the formula or in the denoted object (see the above example on successors and areas). Second, algebraic transformations can produce different expressions holding different algebraic senses, but with the same denotation. For example, transforming n (n+ 1) into n²+n does not change the denotation but does affect its algebraic sense, i.e. the computational rule. Third, it is not always true that two expressions having the same denotation can be mutually reduced by means of algebraic transformations (x+5)² = x and x²+x+1=0 in R. Fourth, algebraic transformations are not always invariant with respect to the denotation (for example, SQRT(x²) and x). Each of these imbalances creates uncertainty in students about the status of the formulas they are using and i s the source of possible misunderstandings. All possible senses of an expression constitute its so called %#(/#+%)#"2 aspects, while its denotation within a universe represents its so called /K(/#+%)#"2 aspect. It is worthwhile observing that the official semantics used in mathematics, and particularly in algebra, cuts off all intensional aspects, insofar as it is based on the assumption of the extensionality axiom (two sets are equal if they contain the same elements, independently from the way they are described or produced), which entails the %#."*%"#B/ of mathematical objects with respect to their intensional aspects. Of course, mathematicians do also use intensional aspects of the mathematical language. but these remain more or less implicit in their definitions and proofs, that is in the way they do mathematics, while their explicit sentences always assume extensionality; this game is very subtle and intriguing and is the cause of many misunderstandings in many pupils, at all ages. In fact, much research has pointed out (see Drouhard (1992). Kieran (1989)), that intensional aspects are very important because it may be very difficult for students to conceive the above invariance. Many algebraic difficulties can be described as

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deficiencies in the way pupils master the invariance of denotation with respect to the sense: there is a sort of rigidity that makes them act as if there were a one-one correspondence between sense, denotation and formal expression, so that identifying all three, they disappear and pupils remain with a trivial denotation: a symbolic expression denotes itself as a collection of signs. They own algebra only as a pure syntax: for them secondary processes, in the terminology of Sfard (1991), are purely syntactic rules and nothing morei, their real meaning, namely Frege’s triangle has collapsed into a trivial object, because of the identification of denotation with sign. As a concrete example of such an identification, let us consider the D+/4') +(*4B(4*"2 students described by Sfard (1992); the point is that they do not realise, in the case of equations, that 'the concept of a truth set the set of numbers which must not change under the ‘permitted’ operations is where the decision to call certain manipulations ‘permitted’ becomes clear' (Sfard (1992), p.2); since they do not see the invariance they become ‘formalist’, in the sense that they reveal a 'basic inability to link algebraic rules to the laws of arithmetic' and so 'formal manipulations ... (remain) as the only source of meaning' (ibid., p.8). Such difficulties in the semantics of algebra can be understood quite well if they are compared with the enormously easier semantics of arithmetic, where there is no imbalance between sense and denotation in a numeric expression; in fact a numeric expression denotes always a definite number and its sense is practically unique; hence, while in an arithmetic expression each time only one interpretation is available, on the contrary an algebraic expression can be generally interpreted in many different ways. —

—

E)#B/D(4"2 7*"9/+

To have a precise description of the dynamics of algebraic thinking, we still need an ingredient, that is the notion of conceptual frame. We shall introduce it by discussing a concrete example: it is a protocol of a typical 'average' student, whom we have called Ann, it is given here as a paradigmatic case. Ann is a 20 year old undergraduate student of mathematics and has to solve the following problem: ?*)./ (,"( (,/ #493/* SDIXTSbiIXTnl %+ "# /./# #493/*1 D*).%'/' D "#' b "*/ )'' D*%9/+: Protocol of Ann Episode 1. Ann develops the formula writing the words /./#1 )'' on the paper near the formulas: (p-l)(q²-1)/8 = (p-1)(q+1)(q-1)/8

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Ann points to the components of the formula and says: o/./#1@/./#1@/./# :::::::::::,999 :::::: (,/ */9"%#%#$ #493/* %+ #)( /./#@::::p Episode 2. Ann makes some algebraic transformations to the formula using the words even, odd: (p- 1)(q²-1)/8 = (pq²-q²-p+1)/8

Ann makes some oral calculations of the type «odd times odd is odd» then says: «...hmmm...it does not work!». Episode 3. As in the previous episode, but with calculations of the type «odd times odd is odd» referred to the factors (p-1), (q²-1); then Ann says «there must be some formula to use for primes! . Episode 4. Ann draws some scribbling on the formulas of the preceding episode and starts verifying the formula with some primes: data are collected into a table:

Ann comments: «So it is already q squared minus one that is a multiple of eight».

Episode 5. Ann changes the sheet of paper and writes down the following:

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Then Ann writes the following formulas:

(2h+1-1)((2k+1)²-1)/8=2h(4k²+4k+1-1/8 = 2h 4k(k+1)/8 and says: «...%7 = %+ /./#1 7)4* S(%9/+T = %+ " 942(%D2/ )7@l (Anne points to the 8 in the formula), so %( */9"%#+ " 942(%D2/ )7 V (Anne points to the 8 in the formula), "#' >/ "*/ h/ "*/@)./*qp:

Episode 6. Ann looks again at the text of the problem and says: o64(@D*%9/+@,"./ $)( #)(,%#$ () ') >%(, (,%+q h'' #493/*+ "*/@/#)4$,p: The example is typical of the way average attaining pupils reason with formulas. High attaining pupils instead, use at once the formula:

(2h+1-1)((2k+1)²-1)/8 = 2h(4k²+4k+ 1-1)/8 = 2h 4k(k+11)/8 and argue directly as follows: «if k is even then we are finished ‘cause four (times) k reduces with eight; if k is odd then (k+1) is even and the same argument applies». Some students also observe connections with triangular numbers. Low attaining pupils instead generally try the transformations of episodes 1-3 or similar ones; they seem to look (more or less in a random. purely syntactical fashion) for some more complicated formula which can solve the problem and seldom try to substitute numbers for letters to see ‘what happens’, so in the end get lost or solve partially the problem. If we look carefully at the dynamics of the protocol. we can observe that the crucial moment consists in the sudden change of strategy from episode 4 to episode 5:

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In fact, all actions and decisions of Ann in the first part (episodes 1,2,3) are ruled by her knowledge concerning even and odd numbers: such knowledge consists of an organised set of notions (i.e., mathematical objects, their properties, typical algorithms to use with them, usual arguing strategies in such a field of knowledge, etc.), which suggests to her how to reason, manipulate formulas, anticipate results while coping with her problem, that is how to switch on senses of formulas to be interpreted and to be manipulated in order to solve the problem. We call such an organised set of knowledge and possible behaviours a B)#B/D(4"2@7*"9/: We take the term 7*"9/ from artificial intelligence studies (for example, see Minsky (1975)); from this point of view, a frame is a structure of data that is able to produce a stereotyped representation of a piece of knowledge. Our notion of frame is wider than that of Minsky, in so far as it entails also specific conceptual aspects of knowledge as an organised set of conceptual notions and operational skills related to some precise pieces of mathematics: so we call it a conceptual frame. As such. it is related also to the notion of B"'*/ (setting), discussed in Douady (1986): its similarity with Douady’s notion rests on the fact that a conceptual frame has also a mathematical dimension (as well as socio-cultural and individual ones). In fact a sense whatsoever can be given to an algebraic expression, insofar as it is related to some piece of mathematics. The notion of B"'*/ in Douady entails a wider mathematical area and is based on mainly conceptual features; our conceptual frame is more specific and limited: a B"'*/ could be that of elementary number theory. a conceptual frame on the contrary must be a specific part of this (for example, evenodd, prime numbers, multiples) and contains not only organised notions and operative tools (as in Douady’s B"'*/T but also precise scripts (condensed stories), with which the subject can operate in an almost automatic way (as in Minsky’s 7*"9/+T: Each conceptual frame entails a certain piece of infomiation and is concerned with what one expects to happen as a consequence of the information and possibly also with what one must do if such expectations are not confirmed. In our example (episodes 1, 2), Ann looks at formulas in order to match a stereotyped sense of parts of) the formula within the conceptual frame (an even times an odd is an even, etc.) with the goal of her task (namely that the product of such parts simplifies by 8). A conceptual frame is a sort of a condensed story, which has its +B*%D(+ and where the subject is supposed to do something, because of the goal of the task. Conceptual frames are activated as virtual texts while interpreting a text, for example of a problem, according to context and circumstances; as such, it has socio-cultural and individual features (the activation of the conceptual frame /./#I)'' by Ann depends on her culture as a student in a certain socio-cultural context and as a specific student with her precise personal history).

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To indicate a wider space, within which a pupil can act and switch on her/his conceptual frames to solve a given task, we use the notion of +)2.%#$ >)*2': With respect to the knowledge domain, where a problem-situation is set, a solving world is featured both by systems of signs, which are used to mediate the subject's thought and action, and by interaction tools, which help the subject to produce meaningful and expressive objects to solve the given problem. For example, Ann does all her performances in a solving world which is featured by the possibility of using algebraic language, numerical tables to represent arithmetic data and natural language, while interacting with a test coacher, who records faithfully what she says. Within her solving world, Ann switches on at least three different conceptual frames: in fact, from the precise conceptual frame and scripts of /./#I)'' #493/*+ (episodes 1,2,3), she passes to the vague conceptual frame and groping scripts of D*%9/ #493/*+ (episodes 3,4), and arrives in the end at the definite conceptual frame and scripts of 942(%D2/+ (episodes 4,5). Of course, such adjectives as precise, vague, definite refer only to the way Ann owns such conceptual frames and stresses the individual dimension of the notion. But let us describe more carefully what happens in the changes from one conceptual frame to the other, in order to understand the dynamics of algebraic reasoning. In episodes 3,4, probably because of her failure to produce the expected result. Ann uses a second conceptual frame, that of D*%9/ #493/*+1 where she shows a less organised operative knowledge and a more groping strategy: this second conceptual frame is less ‘in focus’ than the previous one and remains in the background, while the former continues to rule her actions, even if in a less definite way, as long as her different attempts are unsuccessful. The first phase (up to episode 3), is marked by stereotyped syntactical transformations: the conceptual frame /./#I)'' #493/*+ guides them, but as it is clear from the involution of this phase, semantic control is very feeble; formulas are used as such and do not live really as objects to think with; they only incorporate the frame in a shallow way: the formula continues to be manipulated according to standard stereotypes which belong to Ann’s operative knowledge concerning evenodd numbers. The idea that some magic formula must be used culminates in the end of episode 3, which marks a change of the conceptual frame SD*%9/ #493/*+T and a deep change of approach: now the reasoning becomes arithmetic; semantics is monitored: the conceptual frame, which has strong numerical features, is really inhabited by numerical expressions, which are meaningful for her. As such, they are real (,%#=%#$ ())2+1 which in the long run activate hypothetical reasoning in a new precise conceptual frame S942(%D2/+T1 which overlap partially the first one. In episode 5 Ann can read the old formula with a new eye: it is written in such a way to incorporate the old conceptual frame even-odd and moreover is manipulated

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A%$4*/ V

according to an anticipated thinking related to the new conceptual frame S942(%D2/+T: It is not any longer the way that the formula is written to suggest standard manipulations to seewhat happens, but in order to prove a conjecture, the formula is written and manipulated in a certain way. In other words. in the new conceptual frame, the relationship sense-denotation of the formula does appear as a thinking tool. namely its double face is used dialectically to test a hypothesis: intensional aspects are guided and built by extensional ones and conversely (episode 5 , 1st part). This is the first aspect of what we B"22@7)*942"+@ "+ (,%#=%#$ ())2+: namely when formal manipulations are made in deep connection with denotative aspects, in order to produce a shape in the formula, that incorporates an expected sense, because of a supposed denotation. There is a second way in which formulas can be used as thinking tools. This is illustrated by the second part of episode 5. Here the formula presents some stiffness: the formula incorporates the sense that 4k(k+1) is a multiple of 8 in a transparent way only when k is even. Ann has some trouble with the case when k is odd; the formal aspects force her to simplify in a certain maimer and for a while the (conjectured) denotation of the formula (namely its being even after simplifications) does not cope with the right sense (k+1 is even if k is odd): to grasp this it is not required to manipulate the formula as before. but to activate a new part of the symbolic expression according to its denotation. It is the shift in the denotation

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A%$4*/ [

(even -- > odd) which makes this possible. With respect to the first aspect of formulas as thinking tools, the formal expression no longer changes; it is its sense which changes, insofar as we have looked at its denotation in a new way, shifting from the frame 942(%D2/+ to the /./#I)'' one. In other words, the first way of thinking with a formula is to transform (part of) its intension, manipulating it according to its (supposed) extension (Figure 2); the second way is to discover a new (supposed) intension, without doing formal manipulations (Figure 3), but looking at a new (supposed) extension (in a possibly new conceptual frame). In both cases such changes are activated because the intension and the extension of a formula are embedded in one or more conceptual frames. The second aspect is more evident in solutions given by secondary school students (17 years old) to the following problem: gE,/B= %7 "9)#$ */B("#$2/+ )7 " $%./# "*/" (,/*/ %+ )#/ >,)+/ D/*%9/(/* %+ " 9%#%9491 r4+(%70 0)4* "++/*(%)#+e: A typical average attaining student, whom we shall call Bob, has solved the problem as follows: first he has written the formula S=bh; then he has checked some numerical examples; afterwards, he has written the formula S=xy, has expressed the semiperimeter as x+S/x and then has used the standard machinery of calculus to find the minimum perimeter. (Students who have good marks in mathematics generally have written the last formula earlier and have used derivatives in a standard way.

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Low attaining students have got lost with numerical computations or at most have proved the formula for some specific instantiation of S). It is clear that the rewriting of the formula by Bob marks a change of the conceptual frame, namely from "*/" 7)*942"+ to 74#B(%)# 7)*942"+: The sense changes insofar as the extension of the formula changes because of the shifting of conceptual frames. In summary, our model for describing algebraic thinking is based on the following concepts: (i) Frege’s semiotic triangle; (ii) the notion of conceptual frame. Its dynamics develops with respect to: ( 1) the relationships among signs, senses and denotations of algebraic formulas; (2) the activation of conceptual frames. their mutual relationships and changes from one to the other (within a solving world); In other words, the real dynamic aspects of algebraic processes of thinking can be described properly by observing the way pupils modify their semiotic triangles within a frame or passing from one frame to another. For example, Ann’s thing can be described in the first part of her solution by looking at the way she activates senses for the given formula within the conceptual frame /./#I)''1 whilst in the second part one must look at the new conceptual frames: first, at the way the conceptual frame 942(%D2/+ activates suitable re-naming of variables and syntactic manipulations. so that the formula’s sense can match the goal of the task (that is an expected denotation); second. at the way the interplay between the previous conceptual frame /./*I)'' and the present one S942(%D2/+T suggests to Ann to read a new sense in the old formula in order to overcome the last difficulty (that is the case when k is odd). At this point, it is worthwhile observing that high attaining students seem to use the natural language code only as a framework, curtailing most of the conceptual frames used by low and average students and going directly to the last one. But, when interviewed, they spontaneously make explicit most of the curtailed frames with many details; this seems to confirm that also in their case there may be more than one frame (recall the definition of frame as a virtual text. given above). The presence of frames in solution processes may provide evidence of the fact that students are using a full semantics (namely with sense and denotation) which is generally marked linguistically in some way: for example. Ann writes in words and stresses in her explanations the three successive frames, where she develops her algebraic reasoning; Bob renames variables, when passing from one frame to the other.

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J"9%#$ Now our picture of algebraic thinking has the main ingredients and we can use them to describe what happens when a student solves an algebraic problem; as an exercise, one can look at the different behaviours of students who solve algebraic problems, for example those given in Arzarello (1992d): all of them can be focused clearly by means of our model. Also the detailed description given by Paulo Boero (this volume) of s a n e specific thinking processes, typical of algebraic manipulations, fits very well within our model. Now let us describe with some detail the typical way in which the solution process of an algebraic problem is coached by a pupil. First she/he interprets the text of the problem; in the interpretation activity her/his co-operation is the determinating factor which activates one or more conceptual frames. The way this interpretation activity is concocted is crucial. Subjective aspects determine the very way things are thought and the consequent course of thinking (i.e. anticipating thinking) and of writing (i.e. naming). The result of the interpretation will be another text (maybe written. but also written and spoken), into which the first text has been interpreted, and so on (possibly), interpreting this new text again into a further one. One can easily observe that this activity of repeated interpretations reveals remarkable differences from case to case, even in the same chain of interpretations. but the starting point is crucial for the subsequent development of the whole process. For example, suppose that the problem is standard, so that it reduces to the construction of a symbolic expression E (for example for modelling a situation) or to an interpretation of a symbolic expression (such as the problem of Ann). Generally, E contains some terms: unknowns. parameters or more complex expressions built up using variables and other symbols (specific constants, operations, etc.). The first construction-interpretation of letters is crucial insofar as it entails the very process of #"9%#$1 that is of putting ideas into formulas and this requires a good mastery of the ideographic features of the algebraic language. For example, to introduce or interpret variables (and terms in general) one must choose from among different virtual possibilities (within one or more frames): in fact, a variable (or a parameter or a constant) contains implicitly a form of D*/'%B"(%)# (for example, p may stand for a prime, in a generic or specific way). a term expresses a '/+%$#"(%)# (for example, the perimeter of a rectangle may be indicated with 2p or with 2(b+h)) - see Laborde (1 982) for a detailed discussionii. In the case of algebraic problem solving, the process of naming consists of assigning names to the elements of the problem and is aimed at constructing an algebraic expression able to make explicit the meaning of the problem. It implies the activation of a conceptual frame (at least) and of the dynamic relationship between

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sense and denotation of the algebraic expression within it (for more information, see Arzarello et a1. (1993)). in the naming process the role of constructing and interpreting letters (variables or parameters) is crucial. Letters can be used to give a name to the extra linguistic elements (mathematical or not) which are involved in the problem and to emphasise the relationship among the elements within an algebraic expression. The choice of names to designate objects is strictly linked with the control of the variables being introduced: in this process, one main difficulty, especially for novices, derives from the usual impossibility of sustaining the stream of thought by means of natural language: in fact, algebraic formulas are very seldom a linear record of corresponding sentences in natural language. Not only are relationships expressed in a different way, but as soon as formulas become complex, the algebraic language condenses in a concise and precise way relationships whose expression by means of natural language can be made only vaguely and with considerable difficulty. Let us consider a trivial example, to illustrate this. Problem: 60 9/"#+ )7 " $))' B,)%B/ )7 #"9/+ () '/+%$#"(/ (>) B)#+/B4(%./ )'' #493/*+1 +,)> (,"( (,/%* +49 %+ " 942(%D2/ )7@s:@ Experiments carried out with students, from junior secondary school (11-14 years old) up to University, have revealed similar typologies of errors, namely: 2h+ 1+2k+ 1; 2h+ 1+ 2k+ 1t2 x + y; instead of 2h+ 1+2h+3; moreover, a high percentage of students make only arithmetical checks. The example shows that some students, who seem to grasp the relationships among the elements of the problem using the natural language or the arithmetic code are unable to express them suitably by means of the algebraic code. Note that. already in this simple example, the use of natural language for expressing the problem is not possible (otherwise, one remains vague or must use the language in a very sophisticated and complex way), and this makes things already difficult for a significant proportion of students. More specifically, many students reveal that they are unable to use the algebraic code as a mediator between the identified goals of the problem and the relationships among its elements. Hence they are unable to express the algebraic meaning of the problem in adequate algebraic terms. By ‘algebraic meaning of a problem’ we intend the clarification of the relationships among its elements, when the text of the problem is interpreted according to the rules of the algebraic language. A good mastery of the algebraic code in the construction of an expression includes being able to incorporate the meaning of the problem: according to its goals. From the very beginning, good problem solvers usually have a glimpse of a possible path, and use this implicitly in their first trials at naming; usually they are able to incorporate the relationships among the elements and prefigure

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transformations suitable for reaching the solution. This is not the case for poor problem solvers, who proceed more randomly while choosing and naming variables: their naming is weaker and often superficial, in any case not linked with anticipatory aspects but rather influenced by rigid stereotypes. Sometimes the process of naming is impoverished or blocked when the subject constructs or interprets symbolic expressions in a rigid way, without understanding the complex and flexible relationship between sense and denotation in algebraic expressions. As a consequence, the subject usually does not grasp the potential of the algebraic code, i.e. the possibility of incorporating different properties within the naming process. Such an incapability may be an obstacle lo algebraic reasoning since it inhibits flexibility, which is a basic support of the functions underlying the use of algebraic languageiii.

52$/3*" "+ " $"9/ )7 %#(/*D*/("(%)# The activity of problem solving in elementary algebra can be pictured as playing a $"9/ )7 %#(/*D*/("(%)#P of a text in a semiotic system (for example, a problem in ordinary language) into a text in another system (for example, an equation), or from a text in a system (for example, an algebraic expression) into a text in the same system (for example, another algebraic expression). For example, consider a problem to solve (e.g., a word problem, a conjecture to prove, a phenomenon to model, etc.) and the request of an algebraic solution: the solution is nothing more than an %#(/*D*/(/*1 that is an %#(/*D*/("(%)# and the result of some (*"#+7)*9"(%)# (for example, an expansion or a condensation, see the chapter by Boero in this volume) of the text. with respect to the questions it poses. In fact. the interpreter is useful insofar as it makes it possible to know something more about what is interpreted: in the algebraic case the interpreter is a Frege triangle, produced by means of a suitable interpretation (possibly from another one); during such a process, the frame and even the solving world may change. To give a picture of algebraic thinking as a game of interpretations, let us consider our symbolic expression E (interpreter or to be interpreted): to such an expression is attached a set of Frege’s triangles with one vertex in common (the expression): they constitute all the possible interpretations of E. In this set there may be triangles with different denotations and to each of them there may be attached a cluster of triangles with the side expression-denotation in common, that is triangles with the same denotation (and expression): it is only their sense that changes. When the student starts her/his interpretative activity, by various reasons she/he activates one or more conceptual frames. in a solving world: her/his activity with Frege’s triangle(s) lives within such frame(s) and solving world(s). Once a conceptual frame is active, the student produces as a result of her/his interpretation a text (with some

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A%$4*/ s

sense and denotation among the possible interpretations living in the activated frame), and the process of solution of the problem consists in successive transformations of this text, possibly in the production of completely new texts. each with its own sense and denotation. according to the conceptual frames that are activated; also the transformations are made according to the conceptual frames that are active in that moment. The main goal of the game of interpretation is twofold. On the one hand, the chain of interpreters which can be built as a result of such a game is very fruitful for the learning of algebra, insofar as mental representations of pupils are generally made active by interpreters (think again of episodes 4 and 5 in Ann’s protocol). On the other hand, the game of interpretation helps students in producing metacognition about the processes by which they have produced their algebraic interpreters. The term ‘game’, which we have used for these activities, emphasises that, in a good didactic control of such processes, pupils’ acts of hypothetical thought may be verified and criticised by means of social practice coached by the teacher (i.e., social interaction in the class), or by other forms of interaction (i.e., interactivity with specific software, etc.). Here are some examples of interaction tools and activities

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which can help and sustain pupils to perform their game of interpretations. First, mediating tools which are meaningful and stimulating with respect to their culture (for example, computers, but also other machines: for a discussion in another context, see the Chapter 4 in Biehler et a1. (1994)): these can be useful, since they may facilitate connections between arithmetic practice and algebraic formulas. Second, activities of verbal interpretations of the algebraic expressions produced by pupils: the habit of discussing formulas they have produced and transformed, can stimulate reflections, social interactions and so on. The game of interpretation happens at least at two levels, namely among different frames, within a fixed solving world, or among different solving worlds; for example, among the similar frames activated by Ann in her solving world, or among the very different solving worlds activated by students who use a spreadsheet to construct algebraic formulas. Both may be very useful for algebraic learning. The former type of processes activates pupils’ mental work within the concrete constraints of the solving world and may be useful to overcome its own obstacles and difficulties, because of the shifting from one frame to the other. The latter may allow the subject to restructure totally the way under which she/he looks at the problem situation: concrete facts are always the same: not so for the sense and denotation, which changes radically, because of the possible imbalance between the two worlds (for example, think of the problem of Pythagorean triples solved within algebra or within analytic geometry: see for details Lang (1985)). The repeated construction of interpreters within different solving worlds is unified and ruled by the aim of solving the posed problem situation. or at least of grasping its meaning; of course, the latter docs not automatically ensure that the problem will be solved.

!/#("2 +D"B/+1 /."D)*"(%)# "#' B)#'/#+"(%)# A consequence of the preceding discussion is that in order to achieve a meaningful teaching of algebra, it is necessary to build up didactic situations which stimulate the developing of a fruitful game of interpretations, so as to overcome the typical blocks and difficulties of the algebraic learning, which are underlined in the literature (for example, the blind manipulation of formulas, or the incapability of producing a fruitful comparison of senses in formulas, etc.). To do this, micro-didactics is not enough, in so far as playing a game of interpretation means changing one’s habits and styles of thinking and hence entails long term interventions. made with a systematic new way of approaching algebra in the school. As we have already said, didactic interventions are beyond the aim of this chapter; however, it will be appropriate at this point to discuss some processes, which are crucial for the learning of algebra and can possibly be modified in the long term.

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In fact, to understand something about the nature of blocks met by pupils in their game of interpretation. it is crucial to study these processes, deeply connected with their inner language, where pupils are requested to anticipate and to grasp globally (some portion of) a formula, in order to produce some insight of its different possible shapes and senses. Observations and interviews show that sometimes "#(%B%D"(%#$ (,)4$,( operates on the basis of rich numerical tables, constructed by pupils, possibly using the computer; sometimes it is a sort of hypothetical reasoning made in a mental space which has spatial, temporal and logical features (in interviews, pupils use words, metaphors etc, which mark clearly these features, not all necessarily present in the same subject). Its reality depends on the situation (namely, the problem, the pupils, the negotiation in the class, and so on), it gives real feedback to the pupils, who have built it, and allows them to activate their own conceptual models (see Arzarello (1989) for this notion) according to familiar scripts. The creation of such mental spaces usually helps students to avoid stumbling-blocks and to continue in the game of interpretation; in other words, it facilitates all those flowing processes, typical of problem-solving, by which data can be selected, processed according to one's conceptual models, integrated into new pieces of knowledge, and so on: perhaps it has some connections with the ideal environments in which scientists like Galileo, Faraday. Einstein made their well known mental experiments. In short, a mental space is a sort of fictitious real world which students themselves create, where they can make experiments and develop hypothetical reasoning. Hypothetical reasoning is essential both in general planning, when pupils elaborate possible solution strategies, activating different frames and senses and in controlling, contrasting, comparing them with each other. So it is deeply connected with the symbolic function of language (see the discussion of this notion in Cauty (1984)). As a typical metacognitive skill, it is a long-term process, which can be developed in the long term by means of a cognitive apprenticeship where the teacher encourages the thinking as a stream and not a single act of language. The most difficult point to analyse is that mental spaces are active if (and generally only if) problems are approached from the procedural side, that is doing calculations with concrete numbers, thought of as general objects. namely from a situation where there is a good semantic monitoring (see for example episode 4 in Ann's protocol). However, to get rid of arithmetic and its procedural aspects and to switch to algebra, with its relational and structural properties, pupils must work in mental spaces where objects lose their extra-mathematical and procedural tracks and must translate them into symbolic expressions, which are highly synthetic, ideographic and relational. To do this, a student is required to write concisely and expressively the amount of information of a term, whose complexity and generality cannot easily govern the language of arithmetic. In some sense, the stream of

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thought which sustains her/his computations and arguments contracts and condenses its temporal, spatial and logical features into an act of thought, which grasps the global situation as a whole. Such an inner process happens in a dialectic back and forth movement with the formula (such processes are clearly visible when pupils interact positively with spreadsheets, symbolic manipulators, etc.): there is a sort of converging process, from the formula to the subject and conversely towards the final solution (this has been studied by Gallo in Arzarello & Gallo (1 994) and by Boero in this volume), which in the end culminates in a single act of thought in the pupil’s head and in a general formula on the paper (or on the screen). The former has some similarity with the process of %#(/*#"2%+"(%)# of speech studied both by Piaget and by Vygotsky. The latter has been called B)#'/#+"(%)#P the word is taken from semiology (see Eco (1984), p. 157) and from Freud (1905) (see his analysis of the wits); it has also some connection with the phenomenon of curtailing, described in Krutetski (1976), and with the property of shortening. typical of inner language (see Vygotsky (1934), chapt. 7). As an example, Arzarello calls condensation the process by which the same symbolic expression refers to two different meanings, possibly in two different conceptual frames, or when one shifts from a conceptual frame where she/he is interpreting an expression to another one, adapting the old Frege’s triangle to the new situation (for, example. when Bob renames variables in problem 2. and in doing so changes from an "*/" to a 74#B(%)# frame). Hence, condensation stresses semantic creativity insofar as it may be related to such things as analogies, metaphors, etc.; other examples are discussed in Arzarello (1992d). At the opposite side of condensation we find a typical process of poor algebraic performers, that we call /."D)*"(%)#: Whilst condensation entails a strong semantic control, on the contrary evaporation concerns the dramatic loss of the meaning of symbols met by most pupils when losing the semantic control in algebraic problem solving. Typically, this happens when they cannot any longer express the meaning of mathematical objects and relationships in ordinary language referring to a subject’s actions, to the very processes of their construction and generation and to any other extra mathematical information about them. In such cases they cannot any longer use algebraic language as a record of their processes of thinking, possibly mediating with natural language. This impossibility provokes the transition to the empty semantics of pseudo-structural students, discussed above, namely to a collapsed Frege’s triangle. As such, evaporation is one of the main obstacles in the developing of an algebraic way of thinking and to a non empty use of the algebraic code. It is interesting to observe that both condensation and evaporation obey a principle of economy: in fact both save space and time in the head and on the paper, compared with other more expensive ways of thinking and representing. But while condensation maximises the content of infomation of an expression preserving the

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F. ARZARELLO, L. BAZZlNl AND G: CHlAPPlNl

right senses and denotations by means of condensing them suitably in the expression; evaporation. on the contrary, minimises senses and denotations of an expression trivialising them: the former produces expressions with a high content of information, the latter reduces this content to a minimum. So the two processes are in a duality with each other; both concern the symbolic functions of algebraic language, the former marking a good semantic control, the second its loss. In this delicate point, the role of language may be crucial, particularly as far as the dialectic between inner and outer language can bypass evaporation, provoking instead condensation. Condensation is apparently a sudden phenomenon, which happens on the spot, but it can be grasped properly if one does not look at the learning/teaching of algebra as a sequence of single acts )7 2"#$4"$/ but as a +(*/"9 )7 (,)4$,(1which breaks dramatically with the arithmetic way of thinking, which pupils are acquainted with from elementary school. Condensation marks deeply the passing from a procedural moment to a more abstract and relational one; it appears at once in the strategies of solutions of pupils and, like all processes which happen on the spot, it is very difficult to analyse, because attention is focused on the single acts of language, which reveal only a sudden change. For example, in all the cases we have studied (see Arzarello (1992b)), most pupils who work successfully at problems. even when explicitly asked, are not able to explain what has happened. The only positive observed fact (directly by some of the authors) is that interviews of pupils who ‘have condensed’, are marked by a massive use of complex expressions of ordinary language (for example, conditionals, subordinates etc.), whilst evaporation is at the opposite side (see examples in Arzarello (1992b)). Moreover, in all control classes, where the teaching style does not encourage the use of natural language in problem solving, only pupils with high verbal performances also get good achievements in algebraic problem solving. On the contrary, in classes where the teacher develops in her/his students the habit of using the verbal (spoken and written) code while solving algebraic problems, to discuss different strategies, solutions, etc., then also pupils of middle verbal and mathematical abilities get good scores. Our conjecture is that in such classes the students acquire the habit of working in different solving worlds, of switching from one conceptual frame to the other, of being flexible with respect to possible hypotheses, in other words of making the game of interpretations in a massive way. In fact, the habit of constructing different interpreters may develop in pupils those forms of flexibility of semantic control that can produce condensations, as a synthesis of different conceptual frames and activated solving worlds.

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NOTES 1

2

3

Secondary processes are the typical algebraic process by which computation rules generate abstract objects 'passing to the quotient' (for example: negative numbers from subtraction, by identifying

suitable pairs of numbers; an algebraic number like , from equation x2-2=0, by identifying polynomials modulo division by the irreducible polynomial x2-2). It is interesting to recall that, from a Frege analysis, a name designates an object (possibly an abstract one) and gives also criteria for its identification, that is gives a way for picking it out mentally as an object that has such and such features, which satisfy the given criterion. Designation and predication of algebraic language correspond to such an analysis. For example, let us look now at a typical case of poor algebraic performances, widely discussed in the literature, namely pseudo structural pupils (Sfard ( 1 992)). Roughly speaking, one can say that they act as i f there were a rigid one-one correspondence between expression, from the one side and sensedenotation, collapsed together, from the other; so for a psuedo structural student. transformations are not as in Figures 2. 3 but as in the following Figure:

PSEUDOSTRUCTURAL TRANSFORMATION

A%$4*/ )7 J)(/ [

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DAVID KIRSHNER*

THE STRUCTURAL ALGEBRA OPTION REVISITED

Some of these [school algebra] activities might be described by teachers or other adults as, say, 'expression simplifying,' 'equation solving' or 'problem solving.' Some others might describe them as 'function rewriting,' 'function comparisons,' or 'modeling,' respectively. Others might describe them as operations in and applications of rational or algebraic functions over the rationales or reals. 64( 9)+( +(4'/#(+ +// 2%((2/ 9)*/ (,"# 9"#0 '%77/*/#( (0D/+ )7 *42/+ "3)4( ,)> () >*%(/ "#' */>*%(/ +(*%#$+ )7 2/((/*+ "#' #49/*"2+1 *42/+ (,"( 94+( 3/ */9/93/*/' 7)* (,/ #/K( b4%H )* (/+(: (Kaput, 1995, pp. 71-72) My point of departure is a quote from Kaput's (1995) powerful and oft cited blueprint for algebra education research which relays common assumptions about rule memorising in school algebra that I take issue with in this chapter. My concern is with the possibility for a structural algebra option that is increasingly difficult to maintain in the face of the rule memorisation assumption. Kaput (1995) reaches an inevitable conclusion from the obvious failure of standard algebra curricula: Acts of generalisation and gradual formalisation of the constructed generality must precede work with formalisms -- otherwise the formalisms have no source in student experience. The current wholesale failure of school algebra has shown the inadequacy of attempts to tie the formalisms to students' experience after they have been introduced. It seems that, 'once meaningless, always meaningless.' (pp. 74-75, emphasis added) But algebra as generalisation is antithetical to structural algebra, which by its nature is formal and uninterpreted: The letters are mere 4#'/7%#/' 9"*=+ or 'elements' about which certain postulates are made .... The very point of elementary algebra is simply that it %+ abstract, that is, devoid of any meaning beyond the formal consequences of the postulates laid down for the marks. (Bell, 1936, p. 144)

*

Dept. of Curriculum & Instruction,Louisiana State University, U.S.A.

83 M: ;4(,/*2"#' /( "2: S/'+:T1?/*+D/B(%./+ )# ;B,))2 52$/3*"1 l[\Zl: U VWWX /* 5B"'/9%B ?432%+,/*+: ?*%#(/' %# (,/ J/(,/*2"#'+:

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My argument in this chapter is that learning algebra in a traditional curriculum is not fundamentally a matter of explicitly memorising and applying rules, but of generating and consolidating +43B)$#%(%./ patterns (Hofstadter, 1985). Therefore, it is incorrect to read the failure of current curricula as indicating the bankruptcy of rule-based approaches to algebra. Rather: an explicit curriculum of rule forms has never been tried or tested. The chapter concludes with ideas about how to structure such a curriculum. Before launching into these arguments, let me acknowledge at the start some of the apparent difficulties and contradictions in the position I am advancing. First of all, the assumption that algebra learning (in traditional classrooms) involves rule mastery hardly seems to be open to question or dispute. Textbooks present rules; teachers expostulate on them; and students toil over endless exercises designed to consolidate them. The middle sections of this chapter examine the arguments of research and of common sense, respectively, for rule based learning. Secondly, what is the nature of these +43B)$#%(%./ D"((/*#+ that students are postulated to be appropriating? What kind of epistemological framework do they fit into, and how are they manifest in particular algebraic content? In the chapter I introduce a connectionist framework for pattern matching, and I provide empirical evidence and argument for pattern matching with respect to polynomial parsing. Finally, if, as argued here, fluency with the alphanumeric symbols system of algebra is a matter of subcognitive pattern matching, what can it mean to implement an explicit rule-following approach to algebraic symbol manipulation? Why would we want to? The final section of the chapter outlines a curricular approach to structural algebra, and takes up this apparent contradiction by returning to an epistemological meditation on the nature and function of rational discourse. STRUCTURAL AND REFERENTIAL APPROACHES TO ELEMENTARY ALGEBRA Broadly speaking, there are two approaches that can be taken to meaning making in elementary algebra. The structural approach builds meaning internally from the connections generated within a syntactically constructed system. Referential approaches import meaning into the symbol system from external domains of reference. This is similar to Peacock's (1 833) pure/applied distinction, except that his concern is with the D4*D)+/ of the investigation, whereas mine is with the 9/(,)'P The science of algebra may be considered under two points of view, the one having reference to its principles, and the other to its applications: the first regards its completeness as an independent

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science; the second its usefulness and power as an instrument of investigation and discovery. (p. 185, quoted in Menghini, 1994)

;(*4B(4*"2 "2$/3*": What is characteristic of structural approaches is the synthetic nature of the objects of study. Typically one starts with undefined terms and axioms, and explores the theorems that can be logically deduced from them. The structures developed often are experienced by mathematicians as having a life of their own (Sfard, 1994). Generally speaking, this is the method of pure mathematics. Structural algebra has been motivated by and applied to a wide range of systems, from numerical systems, to geometric symmetries, to matrix forms (Kaput, 1995). Importantly it is not the topic of the study that characterises structural algebra, but the method. For instance, one can study integers by developing and experimenting with patterns. Or one can logically explore a set of axioms that generates a system related to the integers. Only the latter is structural algebra. Whereas a motivating reference domain is common in pure mathematics, from the methodological point of view expressed here it is not essential. Any approach that builds from undefined symbols and explicit rules is structural according to my definition. This would include the traditional school algebra approach were the usual assumptions about rule memorisation correct. The structural approach has been a part of mathematics education since the time of Euclid. Indeed, it is only in the last fifty years in North America that the mantle of deductive mathematics was passed from geometry, which previously was pure Euclid, to algebra; One way to foster an emphasis upon understanding and meaning in the teaching of algebra is through the introduction of instruction in deductive reasoning. The Commission [on Mathematics] is firmly of the opinion that deductive reasoning should be taught in all courses in school mathematics and not in geometry alone. (College Entrance Examination Board, 1959, p. 23) Clearly this is an important legacy for algebra educators to consider. —

M/7/*/#(%"2 "2$/3*" Referential approaches to algebra all share the property that they import meaning to algebraic symbol systems from external domains. These can vary widely from real world situations (Fey, 1989; Nemirovsky & Rubin, 1991; Usiskin & Senk, 1990), to graphs and tables (Confrey, 199 1; Dugdale, 1990; Goldenberg, 1991;

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Kaput, 1987, 1989; Yerushalmy, 1991) , to arithmetic patterns (Linchevski, 1995: Schifter. 1996). As noted above, Kaput’s (1995) dictum that ‘acts of generalisation and gradual formalisation of the constructed generality must precede work with formalisms’ (p. 74) specifies a purely referential vision of algebra. But this also is the trend in almost all of the algebra education literature. Booth ( 1989) puts it this way: Without an understanding of the semantics of algebra, the mere manipulation of symbols becomes a fairly arbitrary exercise in symbol gymnastics, sometimes performed correctly and sometimes not, but in either case with little sense o f purpose. The essential feature of algebraic representation and symbol manipulation, then, is that it should D*)B//' 7*)9 an understanding of the semantics or referential meanings that underlie it. (p. 58) Similarly Thompson ( 1989) sees ‘the most damaging consequence of defining competence 9/*/20 as possession of correct rules is that we fail to look at incompetence as stemming from impoverished conceptualisations of inaterial from which correct rules should have been abstracted’ (p. 138). Part of the push towards referential algebra steins from the poverty of the referential component of usual curricula, which traditionally is restricted to standard word problems. As well, new technologies are pushing the possibilities for referential algebra in exciting new directions (Kaput. 1992; Thompson, 1989). But even taken together, these forces would motivate only calls for an enhanced and expanded referential component to the curriculum. The */D4'%"(%)# of structural algebra (as seen above) testifies to the fact that the failed curriculum of today is taken as proof that structural approaches are not viable. In arguing against this analysis, my objective i s not to develop and promote a new unitary structural curriculum. Rather I seek to enable a curriculum that honors both structural and referential possibilities for meaning making in algebra. INFORMATION PROCESSING ANALYSES OF ALGEBRAIC ERRORS Putting aside for the moment the common sense observation that learning school algebra is the acquisition of rules, there also is a research base consisting of twenty years of information processing analysis of students’ errors that supports this conclusion (Bundy & Welham, 1981; Carry, Lewis, & Bernard, 1980; Davis, 1979; Davis & McKnight, 1979; Matz, 1980; Wagner, Rachlin, & Jensen, 1984). For the most part, this research has concerned itself with identifying and explaining the 9"2 *42/+ (Sleeman, 1984) that characterise students' incomplete mastery of the symbol system (see Table 1 for a sampling). The usual explanation given for these error types is that they reflect a process of overgeneralisation of the

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correct rules (Matz, 1980). This analysis is consistent with constructivist interpretations of students as builders of the knowledge structures that enable their competence. The clear implication of this research program is that competence in symbol manipulation results when the student has developed an appropriate/viable set of rules. In other words. learning algebra is acquiring rules. C"32/ X: E)99)# &**)*+ "#' E)**/B( M42/+

Errors

Correct Rules

It should be noted, the conclusion that learning algebra is acquiring rules is not independent of the epistemological assumptions of the information processing paradigin itself. Information processing psychology understands the mind according to the model of the serial digital computer. The mind/brain, like the computer, is a D,0+%B"2 +093)2 +0+(/9 which can solve problems and manifest general intelligence through the physical manipulation of tokens (or instances) of symbols (Newell & Simon, 1985). The basic model for reasoning in information processing psychology is a formal one in which explicit rules dictate the processes of deriving new data structures from the givens. Indeed, most studies of algebraic skill have "++49/' a rule based explanation as part of the theoretical framework, prior to evaluation of empirical evidence. At the start of their report, Carry, Lewis, and Bernard (1980) assert that students need to have ‘the legal moves of the algebra game’ (p. 2) (such as the distributive law, difference of squares rule, etc.) explicitly represented in memory, Similarly, Matz’s (1980) framework begins with acquisition of a textbook curriculum:

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This proposal idealises an individual's problem-solving behavior as a process employing two components. The first component, the knowledge presumed to precede a new problem, usually takes the form of a rule a student has extracted from a prototype or gotten directly from a textbook. (p. 95) What can we conclude from the observation that research into algebraic symbol skills makes an a priori assumption of rule based learning? In the next section I argue that the agreement of cognitive research with common sense understandings of algebra as ru le based is not a coincidence. Both reflect philosophical assumptions about the human mind that are deeply entrenched in the modern world view of Western culture. EPISTEMOLOGICAL ASSUMPTIONS In the classical era, Plato advanced his philosophy of ideal forms. The sensory world was understood to be illusory; the true forms of things existing in an ideal realm approachable only through the rigors of logic and mathematics. At the start of the modern era, Descartes (1641/1979) confinned the separation of body (including sensory input) from mind through his meditations on existence and certainty. This S'4"2%+( philosophy is the foundation of our culture’s common sense about mentality. As Gardner (1 987) points out, cognitive science (which includes information processing psychology) is a philosophical descendant of Descartes’ dualism: René Descartes is perhaps the prototypical philosophical antecedent of cognitive science .... Mind, in Descartes’s view, is special, central to human existence, basically reliable. The mind stands apart from and operates independently of the human body, a totally different sort of entity. (pp. 50-5 1) How does the dualist position relate to rule based assumptions of cognition? Viewing the mind as a rule based mechanism provides a material explanation of how mind can be complete unto itself, yet still interact with the body to receive information and govern action. Descartes’ unlikely explanation placed the pineal gland as the interface between mind and body. Viewing the mind in analogy to the serial, rule based computer, resolves such speculations. Indeed as Haugland ( 1985) notes, computational (rule based) explanations of cognition solve a variety of perplexing problems that dualist philosophers have struggled with for centuries: (i) the metaphysical problem of mind interacting with matter; (ii) the theoretical problem of explaining the relevance of meanings, without appealing to a question-begging homunculus; and (iii) the methodological issue over the empirical testability ... of 'mentalistic'

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explanations. The computational idea can be seen as slicing through all three dilemmas at a stroke; and this is what gives it [cognitive science], I think, the bulk of its tremendous gut-level appeal. (p. 2) Mathematics has always held a privileged place within the metaphysics of dualism as the prototypical detached mental realm. Thus cognitive scientists and lay persons, alike, are led to understand algebra as a rule based competence, because of the prior assumption that mathematical cognition exists as a separate, self-consistent domain.

E)#(*"%#'%B"(%)#+ As appealing as the rule based gloss of school algebra is, several aspects of our experience with algebra education are discordant with it. First, and most prominent, is the demeanor of algebra students as regards the rule based curriculum. Students' interest is in seeing demonstrations of rule usage, not in hearing explanations of rules. Explanations usually are not welcome. The constant refrain is 'show me how to do it.' The popularity of John Saxon's (1991, 1992) 'incremental approach' is a case in point. Saxon's curriculum features several innovations including mixed practice sets, and reduced explanation and discussion to increase 'time on task.' The Saxon approach is so contrary to the usual espoused beliefs of mathematics educators that, astonishingly, for several years the National Council of Teachers of Mathematics of the United States and Canada refused him advertising space in NCTM periodicals (Hill, 1993). Nevertheless, his approach is well received by students, and relatively successful in comparison to other approaches to competence in standard tasks (Hill, 1993). The homogeneous grouping of exercises (which is the norm in standard textbooks) and the discussion of solutions (widely espoused by mathematics educators) both are vehicles for the mastery of explicit rules and principles. In contrast, Saxon offers students emersion in the domain of algebraic symbol tasks with minimal segmentation of that domain, and with minimal interference of classroom discourse. If the learning of algebra really is mastery of explicitly given rules, why are those explicit rules so unwelcome by students, and why is their absence so well received? A second indication that the rule based assumptions of cognitive psychology and of common sense may be suspect comes from reexamination of the sorts of errors that students typically make as they develop mastery of the symbol system. Examine, again, the error patterns displayed in Table 1. What is striking and so frustrating about these deviations is that they appear not to be based upon logical fallacy, but rather on their perceptual similarity to the correct rules. Thus there is a lurking suspicion that it is perception, not logic, that is operative in students' learning. Pat Thompson ( 1989) expresses the frustration of mathematics educators

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with the current curriculum: ‘students are prone to pushing symbols without engaging their brains’ (p. 138). I would argue that in a very real sense Thompson's observation is correct, if one takes ‘engaging their brains’ to refer to explicit, conscious cognition. CONNECTIONISM In psychology, B)##/B(%)#%+9 provides an alternative to the dualism of information processing theory (Cummins, 1991; St. Julien, 1997). In analogy to the neurology of the brain, connectionism asserts that cognition is parallel and distributed, rather than serial and digital. The primary cognitive functions are pattern matching and associative memory, not logic or rule following. Connectionism notices that the long chains of extended reasoning that serial digital computers do best, are hardest for humans. Things that humans do best, like recognising faces in different situations and from different angles, are the most difficult feats to simulate on serial computers, but the easiest to implement in connectionist architectures (parallel distributed processers, Salomon, 1993). Connectionist psychology posits dramatic redundancy and a superabundance of active elements, in contrast to the neat, linear processes of rule-based systems (Bereiter, 199 1). Information processing psychology and connectionism differ fundamentally with regard to the relationship between the mind and the environment. In the former, the environment needs to be pre-processed into discrete elements, facts, or attributes in order to be utilised by the inforination processor. This is what makes it so consistent with Descartes' dualist vision of mind separated from the body and from experience. In the latter, perception acts directly upon the cognitive system: Experience organises cognition, rather than vice versa. As Hofstadter (1985) has observed, it is those below the threshold of conscious attention that subcognitive processes organise the categories from which intelligence is manifest. The connectionist framework seems, in general terms, to afford the possibility of an alternative account of algebraic symbol skills that is more faithful to our observation as educators that students' work in algebra is non-reflective and patternbased. The purpose of the next section is to show how this framework can inform our understanding of specific cognitive functions in algebra manipulations. Following this we return to the educational dilemma newly formulated: How can we incorporate linear logical processes in a cognitive system that is not architecturally configured for them? —

—

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THE POLYNOMIAL PARSING PROBLEM The challenge for rule based theories of algebraic skill is to demonstrate the specific, definite rules that are used by skillful algebra symbol manipulators. Abstract algebra provides one model for an explicit rule system for elementary algebra. A tenet of abstract algebra is that the operations of addition and multiplication (and by extension of subtraction, division, exponentiation, and they map a pair of elements into a single radical) are binary operations element (Paley & Weichsel, 1966. p. 138). Thus addition maps (5.3) to 8, subtraction maps (7,9) to -2,: etc. From this perspective expressions like 5 + 3 + 1 are ill-defined, unless some convention dictates a binary parse: either (5 + 3) + 1 or 5 + (3 + 1). The fact that these two alternatives yield the same result, 9, is irrelevant to the mathematical principle, as is clear from the case of subtraction, 5 3 - 1, where, the two alternative parses, (5 - 3) - 1 and 5 - (3 - 1), are unequal. There is a standard convention that utilises a familiar hierarchy of operations for parsing expressions that are not fully determined by parentheses. The formulation of this convention presented in Table II comes from Schwartzman (1977). —

-

C"32/ FF: hD/*"(%)# Q%/*"*B,0

Level 1

addition

subtraction

Level 2

multiplication

division

Level 3

exponentiation

radical

;0#("B(%B E)#./#(%)# (a)

Higher level operations are precedent.

(b)

If adjacent operations are of equal level, the operation on the left is precedent.

J)(/: In this classification, Level 3 operations are said to be ,%$,/* than Level 2 operations which in turn are ,%$,/* than Level 1 operations. This version of the syntactic rule leads directly to the usual binary parse of expressions. For example 3x² is parsed as 3(x²) by rule (a), and 5 - 3 - 1 is parsed as ( 5 - 3) - 1 by rule (b). If the rule-based psychology of information processing is correct, then the expert algebraist must have come to internalise this nile (or some alternative), as the basis for competence in the domain of parsing. This claim is not the simple assertion that

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the competent algebraist 'knows' (i.e., can recite) the rule. Rather there must be some internalised representation of this rule incorporated into the cognitive structure that is activated in parsing situations. On the one hand, it is possible to 'know' the rule, but somehow never use it. On the other hand. one may have forgotten, or implicitly acquired, the rule but nonetheless have it as an active part of one's rule structures. Before presenting data that suggest that the basis for competence in parsing does not make use of a rule of the sort presented in Table II (or indeed, of any rule), it is necessary to map out the full range of implications associated with the hypothesis. The first observation is that a full binary parse implies extraordinary complexity, even for relatively simple manipulations. Consider the typical school task requiring collection of like terms for the polynomial 3x - 2y + 2x - 5y. The evidence of introspection suggests that the terms of the polynomial arc relatively autonomous. One can shunt terms around, with suitable care for the signs. But a fully parsed expression provides no such flexibility. In this case, one's mental representation is [(3x - 2y) + 2x] - 5y, and maintaining the full parse entails something like the following: [(3x - 2 y ) + 2x] - 5y = [(3x + -2y) + 2x] - 5y = [3x + (-2y + 2x)] - 5y = [3x + (2x + I2y)u - 5y = [(3x + 2x) + -2y] - 5y = [5x + -2y] - a 0 = [5x + -2y] + -5y = 5x + [-2y + -5y] = 5x + -7y = 5x - 7y. Whereas such a derivation is formally correct mathematics, it seems rather obtuse to presume that people actually process collection of like terms in this way.1 If we are willing to forego the purity of biliary parse, there are alternative logically consistent rule structures that preserve the information processing assumption of rule based intellection. For instance, one might start from the abstract algebraic definition that subtraction merely abbreviates addition of a negated term (Paley & Weichsel, 1966. p. 27), and move from there to dispense entirely with (b) of the syntactic convention presented in Table II. As Asimov (1959) advises, always adding a negative ‘gives us the opportunity to wipe out subtraction altogether’ (p. 32). This move resolves many of the problems associated with polynomial parsing. The parsing ambiguity of 5 - 3 - 1 is unproblematic, because the underlying mental representation. 5 + (-3) + (-1) is protected from inconsistency by the associative law for addition. Similarly the possibility of applying the commutative law for addition to x - y + z to get x - z + y is subverted because the representation of the former expression is x + (-y) + z. . Less obvious but still true is that dispensing with (b) of the syntactic rule in Table II does not lead to logical contradictions for the other operations2. But taking seriously the regimen of a consistent, rule-based conception of algebraic competence demands a certain discipline. If subtraction is to be taken to abbreviate addition of a negated term in this case, it must do so in all cases. Thus one would have to insist that, say, the 'difference of squares' transformation, x2 - x 2 = (x -y)(x + y) is cognitively represented as the 'sum of a square plus the negation of a

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square' transformation. This proposal has no more plausibility (as a cognitive hypothesis) than the complex polynomial simplification discussed above. There seems to be an inherent ambiguity in the representation of subtractio/negation that is not captured in the alternatives provided by consistent rule structures. EVIDENCE There is some evidence that novices do not operate from a logically consistent set of rules or axioms. Cauzinille-Marmeche, Mathieu, and Resnick (1984) have observed a fractured, ad hoc process in eleven-year-old students who were asked to judge the equivalence of expressions like 685 - 492 + 947, 947 + 492 - 685, 947 - 685 + 492, and 947 - 492 + 685. Kieran (1989) reports the findings: When students did not calculate, their most typical response was to make a judgment about equivalence on the basis of rules for transforming expressions. These rules were often incorrect. The most frequent incorrect rules could be interpreted, according to these researchers, as the dropping or adding of constraints on the principles of commutativity or associativity for addition. (pp. 38-39) In general it is eminently unsafe to generalise from novices to experts. However, there is further evidence that this same kind of ad hoc process underlies competence in algebra for the fully competent mathematician too. As part of a quite different study (Kirshner, 1989), a sample consisting of 137 fourth-year engineering students at the University of British Columbia was asked to evaluate 10 expressions for x = 2 (see Table 111) . C"32/ F 2 2 1 C/+( F(/9+1 7)* ;/#%)* &#$%#//*%#$ -#'/*$*"'4"(/+

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Students at this educational level are generally considered to be expert in elementary algebra, and, indeed, only 14 students in the sample did #)( score perfectly (a total of 14 incorrect responses and 1 omitted response). Clearly these errors are a marginal phenomena: however. they are not random. Twelve of the 15 errors (including the missing response) occurred with the trinomial expressions, #6, #8 , and #10 (seven of the errors occurred with #8). In each of these cases the response given (if any) was compatible with the incorrect parse of the expression (e.g., 19 - 4x + 2 = 19 - (4x + 2) = 9; for x = 2). What sorts of explanations are possible for these data? Perhaps these nearly-expert students construct a non-binary representation for polynomials, whereas their more expert cohorts employ a binary representations? Or perhaps for these students subtraction is just subtraction, while for their cohorts it is represented as addition of a negative? The problem with these rule-based analyses is that they propose radically different cognitive structures for students of very high accomplishment in elementary algebra. A third possibility does not require postulating such major anomaly in the cognitive structures of the erring students. Questions #6, #8, and #10 are nonstandard problems i n that usually no more than one of the terms in a polynomial is constant. It could be that the evaluation of polynomials. for competent performers, is instantiated as an ad hoc, perceptually based, left-to-right practice. (Similarly, perceptually based ad hoc practices would protect against other anomalies like applying the commutative law for addition to x - y + z to get x - z + y,) The need for initial focusing (for substitution purposes) on a middle term that is embedded between two constants may have been just sufficiently distracting to override this constraint for this small minority of students. This explanation has the advantage of leaving the syntactic structure of expressions and the representation of subtraction homogeneous for the entire sample, entailing only a slight modification of cognitive structure to explain the errant behavior. The implication, then, is that competence in algebraic skills is not a matter of knowing the rules, so much as of coordinating pattern-based perceptual cues (see also, Kirshner, 1989). TOWARDS A PEDAGOGY FOR STRUCTURAL ALGEBRA The key insight argued for in the preceding section is that competence, or rather fluency, in a domain like algebraic symbol skills does not imply access to the rules that motivate that domain. At some practical level, as educators, we may known this to be true. But our epistemological frame-of-reference does not permit us to honor and act upon that insight. Rather there is deep frustration and discouragement at the failure of our intensive investment in skill development to pay off with a rational, logical apprehension of the domain of algebra.

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The alternative (connectionist-based) epistemology alerts us to the fact that learning always is grounded in perception and pattern matching as embedded in practices, not in abstraction and rule following. Thus there is no reason to expect rationality to emerge from the solitary activity of working exercise sets. But what, then, should educators do to help students apprehend algebra as a logically structured domain? Indeed, what is rationality? Bereiter (1991) offers the following perspective: Harré’s theory about the social nature of rationality (1979, 1984) provides an illuminating way to think about this question and more generally about how the classical rule-based view relates to cognition. When people try to give a retrospective report of their mental processes, what they tend to do instead is provide a O4+(%7%B"(%)# of their actions (Nisbett & Ross, 1980). Rationality, according to Harré, originates in this essentially social process of justification. What we call logical reasoning, and attribute to the workings of the individual mind, is actually a public reconstruction meant to legitimate a conclusion by showing that it can be derived by procedures recognised as valid. (p. 14) It is too extreme to argue that rules play no role in competent performance, but it is an ancillary role informing cognition rather than constituting it. Bereiter continues: Explicit rules may play a part in learning to think, but (as suggested by the long history of failure of instruction in logic to improve thinking) a very limited one. Turning the social process of justification inward amounts to a kind of self-checking. In this process, one might B)#+42( logical rules in the same way that one might consult rules of algebra while solving a mathematical problem or consult an etiquette book when planning a formal dinner. Rules, thus, may play an important role as =#)..2/'$/ (,"( /#(/*+ %#() B)9D4("(%)#+1 but this is a fundamentally different role from the one traditionally conceived by philosophers and cognitive scientists, where *42/+ B)#+(%(4(/ (,/ B"9D4("(%)# "2 "2$)*%(,9+ (,/9+/2./+: (Bereiter 1991, p. 14) Rationality as social legitimisation suggests the outlines for a new pedagogy of structural algebra. If rationality is not inherent to algebraic manipulation skill, it needs to be fostered in the classroom through specialised activities and discourses. It is not enough to explain rules to students and have them practice their use (as in the usual North American curriculum). Students must be in the role of articulating and justifying their rule usage. Rationality accrues from rationalising. It is not inherent in symbol manipulation.

D. KIRSHNER

96

It is worthwhile, at this point, to revisit abstract algebra as a possible source for the rules that might motivate a rational accounting of algebra symbol skills. Indeed, abstract algebra is fundamentally concerned with justification and explanation. But its focus is on axioms and rules for defining extendible algebraic structures. Most of the preliminary theorems are logical niceties like the uniqueness of the additive identity, that have little to do with the derivations involved in school algebra. Furthermore, representational issues related to the parsing of expressions are (rightly) understood to be an independent notational concern outside of the purview of the theory itself. In these respects abstract algebra can be compared to D*/+B*%D(%./ $*"99"* that dictates how a language should be used, with little sensitivity to actual practices of speakers. (In basic linguistic usage prescriptive grammars attempt to impose one model of speech as the norm for a commnity. In contrast to prescriptive grammars which attempt to impose a standard on a community, descriptive grammar is a more scientific attempt to delve into speakers' (implicit) knowledge that enables their actual language competence (Crystal, 1987). The educational track that I recommend is based on a '/+B*%D(%./ $*"99"* (Kirshner, 1987) that attempts to model the actual practices of fluent users of algebraic language.3 This entails an elaboration of the parsing component as well as the usual transformational rules. Indeed, formalising the parsing rules turns out to play a crucial role in providing the discursive grounding for the transformations. An example can give a sense of this role, and of the pedagogical discourse I consider useful. Consider the rule for canceling common factors in a fractional "K expression ( ), so often overgeneralised to common terms = 3K 3 —

) . Now a discourse involving 'factors' and ‘terms’ clearly is 3+ K 3 relevant here. But the defining characterisation for these lexical items lies within the parsing component. To see this we need to refer to the notion of )D/*"(%)# D*/B/'/#B/ described in Table II. Consider the expression 1 + 3x². The hierarchy of operations determines the parse of the expression as { 1 + [3(x²)]}. The most precedent operation (in this case, exponentiation) is the one embedded within the most parentheses, in that if we were to evaluate the expression (given numerical values for all of the variables) it is the most nested operation that would be performed first. The least precedent operation is the least deeply embedded in the expression when fully parsed. The definitions for terms and factors now follow easily: If the least precedent operation is multiplication, the subexpressions it joins are factors; if addition. terms. Thus the above cancellation error for fractions cannot sensibly be discussed outside of an explicit and formal treatment of parsing rules. Trying to talk about terms and factors without this kind of grounded analysis results in the usual illusion of communication that characterises most school algebra discourse. –

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The irony of this approach to symbolic algebra is that although a rational discourse is sought, rationalisation is understood to be a secondary phenomenon, not always easily fitted to the perceptual base for fluent performance. In my judgment, creating rational discourses in mathematics is fundamental to acculturation of students into characteristically mathematical ways of thinking and participating. Great strides in this direction can be taken by attending to the structure of expressions, usually just glossed over in standard curricula. But even then, it is something of a balancing act to maintain a rationalist discourse in the face of a process of competence that essentially is pattern based and non-rational. Thus one probably would want to teach subtraction as addition of a negative rather than assign a binary parse to all expressions, though neither alternative exactly matches the perceptual basis for fluent performance.

SUMMARY The dualist legacy of Descartes gives us a perspective on mind as disembodied, logical, and complete unto itself. Following this legacy through modern information processing psychology to mathematics education gives us grounds to believe that competence in algebraic symbol manipulation entails a logical understanding of algebra. The problem is that the standard curriculum (in North America, at least) which attempts to instill such symbol skills through didactic presentation and individual practice demonstrably fails to result in students' appreciation of the logical, structural underpinnings of the discipline. Rather it leads, for most students, to the dead end of mal-rules that are resistant to logical explanation and that bear an uncanny perceptual resemblance to the correct rules. This failure of the standard curriculum is pushing mathematics education towards strictly empirical versions of algebra in which meanings stem from and are justified through their use in applicative domains like arithmetical pattern and 'real world' situations. The hope that students also can come to participate in mathematics as a formal, logical study is waning. An alternative epistemological framework which understands cognition as a massively parallel pattern-matching process rather than as linear and logical provides for a new understanding of past failures and future prospects. If the mind is not inherently logical, then simply engaging students in symbol manipulation will not necessarily produce a logical instantiation of those skills. Logic needs to be fostered through particular discursive practices. The irony is that abstract algebra doesn't provide a sound basis for such discursive practices. For instance it provides a most unwieldy model for parsing through binary representation of addition and multiplication. It can be likened to a prescriptive grammar which describes how a language +,)42' be spoken with little regard for the actual practices of speakers. A

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descriptive grammar can provide a closer fit to actual practices, although no rule set exactly matches the perceptual basis for fluent performance. But then, engaging student in rationalising their internal processes doesn't mean that those processes actually need to be rational. Logical discourse comes from the social function of rationalising, not from engagement with inherently logical artifacts.

NOTES 1

2

Some theorists (e.g., Ernest, 1987; Drouhard. 1988) have postulated n-ary operators that permit more spontaneous grouping procedures. The arguments here are somewhat technical. Multiplication is associative, so the parse of "3B as S"3TB or "S3BT is unproblematic. Other operations have their parse specified by the +4*7"B/ 7)*9 (Kirshner, 1987) of the operations. For instance " 3 B is ambiguous, but it is arithmetic, rather than "

–

3

"

algebraic, notation. Algebraically one would write either — or — with the parse being determined B 3 –

B

3

by the length of the vinculum. The parse of the other operations are similarly determined by surface form. This is a reinterpretation of Kirshner (1987), reflecting an epistemological turn-around for the author. In Kirshner (1987) the grammar was formulated as an explanatory model displaying the (unconscious) rule structures underlying competence (cf. Chomsky, 1957; 1965).

PAOLO BOERO*

TRANSFORMATION AND ANTICIPATION AS KEY PROCESSES IN ALGEBRAIC PROBLEM SOLVING

INTRODUCTION This chapter aims to deepen the idea that one of the crucial aspects of algebraic problem solving (which might be used to characterise it) is the transformation of the mathematical structure of the problem in order to be able to manage it better, and anticipation which allows the process of transformation to be directed towards simplifying and resolving the task. The process of transformation may happen without, before and/or after algebraic formalisation.When it happens >%(,)4( or 3/7)*/ "2$/3*"%B 7)*9"2%+"(%)#1 it is frequently based on the transformation of the problem situation through arithmetic or geometric or physical manipulation of variables (adding, subtracting, translating, equilibrating...). These problem solving strategies can be called ‘pre-algebraic’. When the transformation happens "7(/* "2$/3*"%B 7)*9"2%+"(%)#1 it is frequently based upon the ‘transformation function’ of the algebraic code. In this case, the manipulation of the algebraic expression extends enormously the range of possibilities of transformation. At least a partial ‘suspension of the original meaning’ of the transformed expression may happen during the transformation process (cf Bednarz & al., 1992). The process of transformation needs specific prerequisites and skills. In the case of transformation after formalisation, a crucial prerequisite is the mastery of standard patterns of transformation. A common ingredient of all the processes of transformation (without, before and/or after formalisation) is "#(%B%D"(%)#: In order to direct the transformation in an efficient way, the subject needs to foresee some aspects of the final shape of the object to be transformed related to the goal to be reached, and some possibilities of transformation. This ‘anticipation’ allows planning and continuous feed-back. In the case of transformations performed after formalisation, anticipation is based on some peculiar properties of the external algebraic representation.

* Dipartimentodi Matematica, Universitàdi Genova, Italy

99 M : ;4(,/*2"#' /( "2: S/'+:T1?/*+D/B(%./+ )# ;B,))2 52$/3*"1 ZZIXXZ: U VWWX /* 5B"'/9%B ?432%+,/*+: ?*%#(/' %# (,/ J/(,/*2"#'+:

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A%$4*/ X: +/9n7)*9 '%"$*"9+

One focus of this chapter is to consider the educational strategies which could enhance the development of the ‘anticipation process’ and the chapter ends with an analysis of some traditional and innovative practices. Examples related to different school levels will be integrated into the presentation, in order to show different aspects of the same topics. AN HEURISTIC MODEL FOR THE TRANSFORMATION PROCESS For heuristic purposes, I will use the kind of diagrams as shown in Fig. 1. In these diagrams, form means any written expression based on the use of the algebraic language; this wide definition covers a great deal of mathematical expressions (eventually integrating special symbols used in different mathematical fields: mathematical analysis. linear algebra, probability....): from arithmetic expressions (such as 3 * [ 2+ 5 * (7+2 * 3)] ) to algebraic equations, from trigonometric equations (such as 1) to differential equations (such as y’(x)=ay(x)-by²(x)), from functional expressions (such as T(ax+by)=aT(x)+bT(y)) to matrix expressions. I will consider +/9 as a mathematical or non-mathematical cultural object (a mathematical statement, a relationship between physical or economical variables, and so on). +/9 consists of a mental representation "#' an external non-algebraic representation. These two expressions are suggested by the classification proposed by C. Janvier (1987. pp. 148-149): ‘the word ‘representation’ has roughly three different acceptations in the psychology literature: at first, ... material organisation of symbols,.which refers to other entities or ‘modelises’ various mental processes ....( see ‘external representation’ in the text above); the second meaning ... refer to a certain organisation of knowledge in the human mental ‘system’ or in the long-term memory... (see ‘mental representation’ in the text above); the third meaning refers to mental images. In fact, it is a special case of the second one’. See also Duval (l995, pp. 25-26) for similar definitions. I would like to point out the fact that the 7)*9 - +/9 distinction, as proposed in this article, does not follow the traditional +0#("K - +/9"#(%B+ distinction. Indeed,

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+/9 brings its own external representation (for instance. a geometric figure - and/or a sentence of natural language). This choice can be justified by the need of analysing some algebraic problem solving processes, especially the activities performed at the 7)*9 level and their relationships with the problem situation ‘represented’ at the +/9 level. For a discussion about possible ‘meanings’ attributed by students to 7)*91 see Demby, 1996; for an in-depth study of some, delicate questions related to the 7)*9 - +/9 distinction, see Arzarello, Bazzini & Chiappini, 1994 and Chapter 4 of this volume. In 7)*9n+/9 diagrams, upward arrows mean ‘formalisation’, downward arrows mean ‘interpretation’. Formalisation consists of a translation from +/9 into an expression of the algebraic language. Interpretation means generating a mental representation and an external lion-algebraic representation coherent with 7)*9: Continuous horizontal arrows mean ‘transformation’; or more precisely: horizontal arrows between 7)*9 X and 7)*9 V mean ‘transformation according to the rules of the algebraic language’, including not only standard algebraic transformations of a literal expression, but also resolution of differential equations, of systems of linear equations, etc.; also included are substitutions of numerical values to letters. 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 original one. The following are some examples illustrating the above ideas: i)

transformation from: (a4-b4)/(a+b) to: a 3 - a 2 b + ab 2 - b 3 can be performed through decomposition: a4 - b4 = (a-b)(a3 - a2b + ab2 - b3) and simplification;

ii) transformation from: (sinx exp2x)' to: (cosx + 2sinx)exp2x can be performed according to the theorem concerning the derivative of a product, and application of the distributive property: iii) transformation from: y''(x)+4y(x)=1 to: y(x)=A sin2x + B cos2x + 1/4 can be performed through standard methods of resolution of linear differential equations. - horizontal continuous arrows between sem 1 and sem 2 mean ‘transformation of

mental and corresponding external representations’. The example concerning the evaluation of the area of a rectangular trapezium, illustrated on page 103, shows how this transformation can be performed in this case (through a change of the decomposition of the trapezium).

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I observe that 7)*9 X may be equivalent (through reversible algebraic transformations’) to 7)*9 V , and +/9 X may be equivalent to +/9 V (same example quoted above). - horizontal dotted arrows indicate a ‘guess’ (conjecture to be proved, etc.). I will consider now some examples of usage of the heuristic model I have introduced. These examples will show how some algebraic problem solving activities can be schematised through the model and prepare the analyses performed in the following sections. The complexity of mental operations involved in algebraic problem solving and revealed through the +/9 - 7)*9 diagrams suggests some, possible reasons for the difficulties met by students.

5DD20%#$ " 7)*942" () +)2./ " 9"(,/9"(%B"2 )* #)# 9"(,/9"(%B"2 +("#'"*' D*)32/9: 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 produce a ‘result’ form 2; the interpretation of form 2 produces a new ‘meaning’, sem 2. In many cases, this process is a multistep process (with a chain of fundamental diagrams of the type considered before). #$%&'()* it is well known that the ‘stopping distance’ s of a car, from the point where the driver sees the danger, can be determined by adding a distance proportional to the square of the speed v to a distance proportional to the speed v (depending on the quickness of reflex). Let us consider the problem of determining the range of the speed which is compatible with the ‘stopping distance’ of 100 m; we may put the law stated before (sem 1 ) into a formula; we then get, for 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 (particularisation of the situation, bringing to sem 2 and: correspondingly, form 2 ). For a common situation (modern cars, normal conditions of the road, mean reflex speed) we may pose: A= 0.006; B= 0.08 (if v is expressed in km/h, and s in metres). So, we may solve the inequality : s = 0.006 v ² + 0.08 v = 100 (form 2). We get (through standard formulas): -136 = v = 123 (form 3). Interpreting this result, we may say that the speed must not exceed 123km/h(sem 3). The following diagram synthesises the whole process:

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103

We may observe that the root -136 (obtained through the resolution of the equation) is not relevant to our problem; this shows the importance of the ‘interpretation’ phase of the ‘algebraic result’ form 3. ?*)'4B%#$ #/> =#)>2/'$/ "3)4( "# )D/# D*)32/9 +%(4"(%)#: Suitable transformations at the sem level and/or at the form level can produce new knowledge. The new knowledge may concern: - a conjecture about the existence of a transformation between 7)*9 X and 7)*9 V, suggested by relationships existing between +/9 X and +/9 V This is illustrated by a simple example concerning the evaluation of the area of a rectangular trapezium:

7)*9 F = ",nV + 3,nV

7)*9 V = ", + S3I"T,nV

In this case, two different decompositions of the original figure into simpler figures generate two different formulas; but sem 1 is equivalent to sem 2, and this suggests that a transformation may exist between form 1 and form2 .

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the existence of an ‘object’ related to +/9 X1 whose existence is a consequence of the interpretation )7 7)*9 V, derived from form X according to more or less standard transformations This is illustrated by an )$%&'() suggested by Paolo Guidoni: it is not difficult to verify through measure that the equilibrium temperature T f reached by the mixture of two quantities of water, m 1 , and m2 is related to their respective temperatures T1 and T2 at the moment of the mixture according to the –

following formula (form 1 ):

This formula may be interpreted as: ‘Tf is the weighted mean of the temperatures T1 and T2’ (sem 1)

By a very easy algebraic transformation we may write :

This formula (form 2) may be interpreted as ‘conservation of the quantity of heat’ (sem 2). By a suitable algebraic transformation of this formula we may write the following formula:

This formula may be interpreted as ‘inverse proportionality between the quantities of water and the absolute variations of temperatures’(sem3). The following diagram synthesises the whole process:

KEY PROCESSES IN ALGEBRAIC PROBLEM SOLVING

105

Some of the more spectacular applications of mathematics to physics concern this kind of usage of mathematics; physicists ‘put a physical situation S+/9 XT into a formula’ S7)*9XT (an algebraic formula, a differential equation, etc.); suitable (more or less standard) transformations of the formula may generate a new ‘formula’ S7)*9 2), the interpretation of which S+/9 VT may increase our knowledge of the physical world. Another remark: a kind of principle of ‘neutrality’(in relation with the real world) of algebraic transformations (already realised by Galilei) allows us to operate in such a way, that if an hypothesis +/9 X is appropriately put into a formula 7)*9 X 1 the interpretation of a transformedform V formula (obtained from form XT may be used as a tool to validate sem X: ?*).%#$ " B)#O/B(4*/ S%# 9"(,/9"(%B+1 D,0+%B+ "#' +) )#T: Algebraic formalism is a current tool in proving conjectures. In this case, frequently, sem 2 is known (the ‘content’ of the conjecture), sem 1 is known (information about data: physical situation, relationships between variables), form 1 and form 2 must be expressed in a convenient way in order to get form 2 starting from form 1 with suitable transformations.

In many situations the passage from 7)*9 X to form V (and, consequently, from +/9 X to +/9 V) needs intermediate steps, according to a chain which may be more

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or less complex. theorem:

As a very simple example, we may consider the following

‘The sum of two consecutive odd numbers is a multiple of 4’ Through a suitable formalisation, we may write the sum of two consecutive odd numbers as: (2K + 1) + ( 2K + 3); performing standard algebraic transformations we get: ( 2 K + 1 ) + ( 2 K + 3 ) = 2K+1 + 2 K + 3 = 4K+4= 4 ( K + 1 ) ; the interpretation of this formula allows the validation of the thesis.

TRANSFORMATION BEFORE ALGEBRAIC FORMALISATION Some transformations of the problem situation, having a counterpart at the form level, can be performed without any algebraic formalism. This subject was investigated in collaboration with Lora Shapiro. First of all we recall some essential contents of our research report (Boero & Shapiro, 1992). The purpose of this study was to understand better the mental processes ( i.e. planning activities, management of memory, etc.), underlying students’ problem solving strategies in a ‘complex’ situation. To this end the following problem was administered to students from grade IV to grade VIII: ‘With T liras for stamps one may mail a letter weighing no more than M grams. Maria has an envelope 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 have been proposed to different groups of pupils:

The students’ resolution strategies have been analysed according to a classification scheme suggested by the data from a pilot study, and corresponding to the aim of exploring the mental processes underlying these strategies.

KEY PROCESSES IN ALGEBRAIC PROBLEM SOLVING

107

Strategies were coded in the following manner: ‘Pre-algebraic’ strategies (PRE-ALG.). In this category the strategies involved taking the maximum admissible weight and subtracting the weight of the envelop from it. The number of sheets is then found by 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 algebraic form, these strategies would correspond to transformations of the form : –

Sx + E

to :

up to :

x = (M - E)/ S

For the purposes of this research, we have adopted the denomination ‘prealgebraic’ in order to emphasise two important, strictly connected aspects of algebraic reasoning, namely the transformation 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. ‘Envelope 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 envelope 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 envelope; ‘trial and error’ strategies in which the solution is reached by a succession of numerical trials, keeping into account the results of the prceeding 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 envelope, and thus hypothesises that the maximum allowable weight is filled by sheets, and then decreases the number of sheets by one, etc. A preliminary review of the results (see Boero & Shapiro, 1992) showed that there is a clear evolution with respect to age and 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 grade to V grade; and from VI grade to VIII grade ). We see that the motivations and access to prealgebraic strategies may be different; but in all of them there is a form of reasoning that may derive from a wide experience involving production of ‘anticipatory thinking’. That is to say, with the aim of economising efforts, pupils plan 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 –

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grade IV to grade V and in grade VI to grade VIII, as well as with respect to the results involving more difficult numerical data (in the case (250, 14, 16), results show an higher percentage of PRE - ALG. strategies at every age level). Concerning research findings in the domain of pre-algebraic thinking, we may observe that there is some coherence between: our results, concerning the influence of numerical data on strategies in 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 ( 1991), concerning numerical equations proposed to seventh graders prior to any instruction in the domain of equations. For instance, they find that the equation 4n + 17 = 65 is solved by 41% of seventh graders by performing 4n= 65-17=48 and then n= 4814, while the equation 13n + 196 = 391 is solved in a similar way by 77% of seventh graders. Taking into account the Herscovics & Linchewski’s (199 1) and Filloy & Rojano’s (1989) findings, we have performed a further analysis of our data which gives evidence of two extreme opposite patterns, and many intermediate behaviours of pupils engaging in a PRE-ALG. strategy. Some students seem to transform the problem situation by thinking about the number of sheets and the weight of the envelope as physical variables; indeed 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 on 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 understand better the degree to which different approaches to the 'transforming function' of the algebraic language are complementary. Our study gives some information about the B)$#%(%./ *))(+ of algebraic transformations. As we saw in the preceding paragraph, algebraic transformations (especially the more open and complex ones) require the student to integrate two or more of the following activities: transforming the nature of the problem (through horizontal and vertical arrows), in 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 expressions and/or on the variables, and/or through the formalisation process ) making choices in order to obtain the solution in an economic way: suspending the original reference meaning (at the sem level) of algebraic expressions, and working at the level of algebraic transformations; –

–

–

–

–

–

KEY PROCESSES IN ALGEBRAIC PROBLEM SOLVING

109

using the reference to the meaning (at the sem level) to plan further steps of transformation of form (cfr. Radford, 1994: ‘semantic deduction’), or to interpret the consequences of performed transformations. If we consider the ‘sheets and envelope’ problem and the resolutions achieved by students, we realise that (depending on age and instruction) many of them, while producing and managing PRE-ALG. strategies, were able to integrate some of these activities in an effective way. –

DEEPENING THE TRANSFORMATION FUNCTION OF THE ALGEBRAIC LANGUAGE

We observe that any algebraic expression may be transformed into different expressions, and any transformation may be achieved through different patterns, according to different aims and criteria. I will try to explore some aspects of this ‘transformation’ process related to its aims and components. To do this, I will start by analysing an example in some detail: #$%&'()* this is the case of trigonometric equations deriving from mechanics or geometry problems; the transformation process is performed in order to bring them to a well known, easy to process expression: 2 2 2 2 sin @x + cos2 2x = 3/2 becomes: sin x + (cos x- sin x ) = 3/2 , 2 and then: sin 2 x + (1 - 2sin2 x ) =3/2 ,.............., 4 2 and, finally, 4 sin x - 3 sin x-1/2 = 0, which is easy to solve through the substitution: sin 2x = y . As regards the next points i), ii), I observe that the (standard) transformation of 2 2 B)+ 2x in terms of +%# K is suggested by a guess concerning the possibility of 2 writing down an equation in the ‘unknown’ +%# K : in the case of a high school student familiar with trigonometric equations, the experience gained in similar situations and the initial shape of the equation allow a transformation suitable to facilitate the task of solving the (transformed) equation. The transformation process, guided by this intuition, is performed according to standard patterns. Taking into account the analysis performed in the preceding example, we may consider the following working hypotheses:

+, The transformation function is performed through a -+%()./+. 0)(%/+1234+' between 3/%2-%0- '%//)023 15 /0%23510&%/+12, deriving from instruction and practice, which produce the transformations, (for instance, considering the equality: 2

2

b -a = (b-a)(b+a);

or: a/b+c/d=(ad+bc)/bd; or: (fg)'=fg+fg') ) and anticipations which suggest a suitable ‘shape’ for the formula to be processed and the direction of transformations. Concerning the words utilised to express this working hypothesis, I observe that the word ‘anticipation’ means the mental process through which the subject foresees

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the final (and/or some intermediate) shape of an algebraic expression useful for solving the problem, and the general direction of the transformations needed to get it. Different elements may be concurrent in this process: the memory of past, successful transformations performed in similar situations (i.e., experience); the intuition of possible, final or intermediate shapes of the algebraic expression, suggested by its present shape; the capacity of relating the shape of a possible transformed expression to the aim of solving the problem. With the word ‘dialectic’ I want to emphasise the fact that if the subject has the necessary prerequisites and experience to attack a problem needing algebraic transformations, his success depends on a functional dynamic relationship between the two ‘poles’ (standard pattern of transformation and anticipation) whose characteristics are different and, in some senses, opposite. The continuous tension between ‘foreseeing’ and ‘applying’, ‘guessing’ and ‘testing the effectiveness’ allows the productive development of the process of algebraic transformation. In general, standard patterns of transformation without anticipation offer blind perspectives - with the exception, for expert people, of some easy school exercises on ‘simplification’ of algebraic expressions or standard resolutions of equations. Concerning the expression ‘blind perspectives’, here are two very simple examples: (example, grade VIII): the student is requested to generalise, in the case of the sum of four subsequent odd numbers, the property according to which the sum of two consectutive odd numbers is a multiple of 4. He immediately writes down: –

p + 1 + p + 3 + p + 5 + p + 7=4p + 16 (the choice of the letter p probably depends on the first letter of ‘pari’, which means ‘even’ in Italian ), then he stops: he does not anticipate the divisibility by 8; the probable, original meaning of p (‘even’) and the divisibility of p by 2 remain hidden; at first glance the student only finds the divisibility by 4; later on he writes:

4p + 16 = 4p + 8 + 8 = (4p + 8) + 8, then he stops again; (example, first university year): the student already knows the proof according to which if f and g are derivable functions, then fg is derivable and (fg)'= f 'g+fg';the student must find out what happens with 1/f (if f is a derivable positive function), The student writes down the ‘incremental ratio’ of 1/f at point x: –

(1/f(x+h)-1/f(x))/h=((f(x)-(x+h))/f(x+h)f(x))/h; then he stops: no relationship is recognised with known derivatives, no connection is made with a formula to be proved.

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On the contrary, anticipation may suggest the final expression of the transformed formula or intermediate steps, but - apart from very easy problems - such results cannot be obtained without a sufficient mastery of the standard patterns of transformation. Here are two simple examples: – (example, grade VIII): through examples, students have discovered that the sum of two subsequent odd numbers is divisible by 4; the teacher helped them to write down the initial and final step of the proof of this conjecture: 2k + 1+ 2k + 3 = C * 4 (where C is a suitable number depending on k)

A student writes: 2k + 1 + 2k + 3 = 2k + 2k + 4, then he stops and says: ‘I see, 4 seems to be there ... But I cannot figure it out in the formula’. The standard transformations 2k+2k = 4k and 4k+4=(K+1) * 4 seem to be out of reach of this student. Another student writes: 2k + 1 + 2k + 3 = 2k + 2k + 4 = 4k + 4; then he says: ‘how can it be proved that 4k + 4 is a multiple of 4? ‘.The standard transformation 4k+4=(K+1) * 4 seems to be out of reach of this student, although he knew the distributive property for numbers, as we realised from a previous interview. (example, first university year): students must prove that ‘if f is derivable and 2 positive, then l/f is derivable and (1/f)’ =- f/f ‘; a student writes down the same expression already considered in the previous example concerning derivation, then 2 he says: ‘Oh, yes, I see: f(x+h)f(x) approaches f (x) when h approaches 0.... oh, yes, f(x)-f(x+h) is like - (f(x+h)-f(x)) .... but how can I bring h under the difference (f(x+h)-f(x)) ? The place of h is occupied by f(x+h)f(x)!’ –

++, - such -+%()./+. 0)(%/+1234+' may have different characteristics and develop in different ways in different problems, as shown in the examples under i) and in the following two extreme cases: proving a conjecture (see 2.3.): frequently, the ‘shape’ of the final formula may be easily determined - or it is given; a convenient algebraic representation of the relationship between data must be constructed in order to facilitate the process of transformation towards the final formula, anticipating some aspects of this process, and standard patterns of transformation must be applied to achieve the transformations; constructing a conjecture (see 2.2., second example): the final formula is unknown; exploring, anticipating, transforming algebraic expressions must take place, based on generalising and synthesising numerical experiments and/or establishing algebraic relationships between the variables involved; –

–

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+++, - such -+%()./+. 0)(%/+1234+' needs support by algebraic, external representations (see Janvier, 1987) with suitable characteristics in order to manage ‘patterns of transformation’ and ‘anticipations’ (symmetry, references of literal signs to their meanings,.....). In particular, as we have seen in the preceding examples, and we will see later in more detail, sometimes the shape of the written algebraic expression may provide hints for the process of transformation (thus supporting anticipation), sometimes the shape of the written algebraic expression is suggested by the guess of a possible transfomation suitable for solving the problem. According to our experience, the shape of the algebraic expression, autonomously written by students (or suggested to them by the teacher) in order to solve a problem, has a very strong influence on their performance. As an )$%&'()6 few eighth graders are able to prove that the sum of two consecutive odd numbers is a multiple of 4 (see 2.3.) if the teacher suggests writing two consecutive odd numbers as d, d+2 or p+1, p+3. On the contrary if the teacher suggests taking into account that an odd number may be written as 2k +1, then proof becomes accessible to many students. Another )$%&'()* in order to prove that (p-1)(q 2 - 1) is divisible by 16 if p and q are odd numbers, frequently high school or university students write p=2m+1 and q=2n+1 and finally get (by standard transformations) the following expression: (2m-2)(2n+2)2n= 8(m-1)(n+1)n (cf Arzarello, Bazzini & Chiappini, 1994). At this point, if the teacher does not intervene, many students abandon this track because they do not ‘see’ that (n+1)n is the product of an even number and an odd number ! The presence of m-1 acts as a distracter, the shape (n+1)n hides the existence of an even number in this product! These working hypotheses, which offer a ‘way of viewing’ the process of transformation of algebraic expressions, have been used to plan some experiments with students. Collected data will allow some cognitive aspects of the transformation function of the algebraic language (see 5.), and some educational problems concerning it (see 6.) to be analysed. Thus, it will be possible to understand if the ‘way of viewing’ realised through the previous hypotheses provides some insight into the process of transformation; it will also be possible to deepen the meaning of these hypotheses. COGNITIVE ASPECTS OF ALGEBRAIC TRANSFORMATIONS: THE PROCESS OF ANTICIPATION AND THE ROLE OF EXTERNAL REPRESENTATION From the cognitive point of view, I will try to deepen the role of suitable written algebraic representations in enhancing the previously mentioned dialectic

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relationship or preventing it from taking place. Concerning this issue, two experiments were realised in 1994/95. The 5+03/ )$')0+&)2/ concerned university students with a wide, common university background in algebraic transformations (fourth year mathematics students). The aim of this experiment was to explore the dependence of the two poles of that dialectic relationship (standard patterns of transformation; anticipation) on the possibility of writing algebraic expressions. Do written algebraic expressions. when they are allowed, enhance standard patterns of transformation and/or anticipation? How do limitations in using written algebraic expression affect standard patterns of transformation and/or anticipation? This experiment consisted of proposing different tasks (proving a conjecture; constructing a conjecture) >%(,)4( "#0 */+(*%B(%)#+ for one group of students (group A), and >%(, (,/@7)22)>%#$@*/+(*%B(%)#+ for the other parallel group (group B): when ‘proving a conjecture’, only the ‘final’ and the ‘initial’ algebraic expressions may be written; the proof must be written verbally without using algebraic signs. when ‘constructing a conjecture’, all the explorations had to be expressed verbally; no algebraic sign was allowed neither in the explorations nor in the expression of the conjecture. Here are the conjectures to be proved: (C1) (following an idea by Arzarello): prove that the number (p-1)(q2 - 1)/8 is an even number, when p and q are odd numbers (C2) if K is a natural number, prove that the sum of 2K consecutive odd numbers is a multiple of 4K The conjecture to be constructed concerned possible generalisations, different from (C2), of the property: ‘The sum of two consecutive odd numbers is a multiple of 4’ The conclusion of the analysis of collected data may be summarised as follows: for a very simple initial expression (conjecture C1), anticipation and standard patterns seem to be inore easily developed by the students of group B in comparison with the students of group A (having no restriction in writing the intermediate steps): the possibility of intermediate steps seems to reduce the engagement in planning activities; in the case of a more complex task (conjecture C2), or more general conjectures constructed, both anticipation and recourse to standard patterns of transformation are substantially enhanced by the possibility of managing and transforming written expressions in a written form, even if the two processes seem of very different nature (see the later discussion of: ‘externalisation’ and ‘internalisation’). It is also interesting to see that students of group B produce only simple conjectures, compared to those produced by the students of group A ; again, in the case of the conjecture C2, the choices of the initial expressions (when produced) are more carefully made by the students of group B with suitable letters –

–

–

–

–

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and ‘shapes’ : the prevention from transforming (in written form) the algebraic expressions seems to enhance anticipation; this puts into evidence the nature of the planning process, which is inherent in writing down the starting expression. All this seems to be very strictly related to our observations concerning the role of external representation in problem solving (see Ferrari, 1992): good problem solvers orient the external representation of the problem situation towards its resolution; this means that, from its very beginning, external representation is ‘solving process oriented’, with a balanced relationship between ‘externalisation’ (external realisation

of shapes and steps anticipated in the mind) and ‘internalisation’ (taking and exploiting products of one’s own external actions, or products of other people). The following excerpt, concerning a group A student who tries to prove C2, shows mean in the case of algebraic what ‘externalisation’ and ‘internalisation’ expressions:

(after 2 minutes):

SJhC&PF "9 #)( +4*/ "3)4( (,/ 2"+( (/*9 )7 (,/ +49 Q)>/./*1 >/ >%22 +//T: (after 3 minutes):

(**) ?/*,"D+1 F B"# 3"2"#B/q

F( %+ #)( 0/( '%.%+%32/ 30 4K. N/( 4+ (*0 >%(, +)9/ ."24/+ )7 K K = 1 9/"#+P (>) )''1 B)#+/B4(%./ #493/*+ : 2m + 1 + 2m+1 F( ')/+ #)( >)*=q C,/0 "*/ #)( B)#+/B4(%./q Perhaps, the first is 2m +1 and the last is V9 t V,/# (,/0 "*/ ()$/(,/*1 %+ " >"0 () +%9D2%70 942(%D2%B"(%)# ’. I: g5#' (,/#1 m,"( >)42'@0)4@')xe S: g[Z '%.%'/' by X[ ‘. S-+%#$ B"2B42"()*T gF( >)42' 3/ [e: I: g5#' %7 0)4 >)42' 4#'/*+("#' %( %# (,/ +/B)#' >"0xe S: gX[ 2)(+ ) 7 K #493/* gives [Ze:

I: gQ)> ') 0)4 +)2./ %(:xe S: gF ')#e( =#)> ,)> () B"**0 %( ) 4 ( e @

I: gN/(e+ >*%(/ %(e: S : Sm*%(/+TP gC,%*(//# 2)(+ ) 7 K %+ $)%#$ () 3/ [Ze: gI 4#'/*+("#' (,%+ >"0P K %+ "# 4#=#)># #493/* >,%B, B"# 3/ 7)4#' >%(, "# /b4"(%)#: C,/ )(,/* 7)*9 F 4#'/*+("#' %( %+ S+,/ >*%(/+TP C,%*(//# 942(%D2%/' 30 K %+ $)%#$ () 3/ [Z ’. I: g5#' (,%+ gSC,%*(//# 30 K %+ $)%#$@()@,/ [Z eT >"+ +)2./' ')%#$ >,"(xe S : gL%.%+%)#e: I:

g5#' 0)4 9"'/ +4*/ %( >"+ B)**/B(1 ,)>x

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S : S-+%#$ B"2B42"()*Te2[ 30 [ /b4"2+ [Ze:

I: gC,"( %+1 0)4 =#)> ,)> () +)2./ (,%+ SgC,%*(//# 30 x %+ $)%#$ () 3/ [ZeT1 34( SgX[ 2)(+ )7 K #493/* $%./+ [Z ’), ')#e(@0)4@=#)> "(@(,%+ 9)9/#(xe S gJ) e: The above text illustrates that the student thinks that only the first use of language can be represented by the algebraic equation 13$=39. In the second use, she thinks of 13$ as a repeated addition. This fact impedes the inversion process, that is 39 divided by 13. This is the reason why she can not solve the equation In her second use of language. In the L/ J49/*%+ L"(%+1 transposition of unknown terms to the other side of the equation does not appear, due to the lack of symbolic language. This allows us to make plausible hypotheses, which can be tested within empirical studies. So, the analysis of texts that predate Viète’s 5#"20(%B 5*(1 together with the development of experimental teaching sequences, suggested to the authors the existence of a didactic cut in the child‘s evolutionary line of thought from arithmetic to algebra. This cut corresponds to the major changes that took place in the history of algebra in connection with the symbolic representation of the ‘unknown’ and the possibility of ‘operating on the unknown’. In terms of the curriculum, the cut is located at the transition between: (‘6/7)*/ ’) The students know how to solve arithmetical equations of the type A (Bx

C) = D

x/A=B x/ A = B/C

In order to solve these it is sufficient to invert or to ‘undo’ the indicated operations. F( %+ #)( #/B/++"*0 () )D/*"(/ )# )* >%(, (,/ 4#=#)>#: (‘57(/*’) Students have received no instruction on how to solve equations of the types. Ax

B = Cx D

To solve these it is not enough to invert the indicated operations. F( %+ #/B/++"*0 () )D/*"(/ )# >,"( %+ */D*/+/#(/':2 Whether ‘the resistance to operating on the unknown’ is or is not an epistemological obstacle in the sense of Bachelard may be the subject of discussion by epistemologists or historians of mathematics. But from the point of view of mathematical education, this analysis of historical texts suggests the possibility of

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studying empirically analogous obstacles within the problem solving approaches of beginning algebra pupils. We will not describe further details of this research. What we want to point out is that the theoretical results of the analysis of classical texts suggested a possible focus for empirical study as well as plausible categories for the theoretical framework of that study. On the other hand, once the results of the empirical work were known, the reading of the historical texts became deeper. Some things that in the first readings appeared as contingent and idiosyncratic, were later recognised as belonging to the way in which specific knowledge is constructed. THE STATUS OF NEGATIVE NUMBERS IN THE RESOLUTION OF EQUATIONS (GALLARDO & ROJANO, 1994).

As individuals acquire algebraic language, the extension of the numerical domain from natural numbers to integers becomes a crucial element for achieving algebraic competence in the solution of problems and equations. This statement led to the design and execution of this project. The didactic study of this research shows that negative numbers are interpreted by secondary school students in various ways: subtrahend, where the notion of number is subordinated to the magnitude, (for example. zero is considered less than negatives); signed number, when a plus or minus sign is associated with the number: relative number (or directed number), where the idea of opposite quantities in relation to a quality arises in the discrete domain and the idea of symmetry appears in the continuous domain; isolated number where there are two levels, that of the result of an operation or as the solution to a problem or equation. Finally, the formal mathematical concept of negative number is reached, within an enlarged concept of number embracing both positive and negative numbers. We intend to show that historical avoidances and recognitions of negative numbers, have a counterpart in different stages of conceptualisation of these numbers manifested by present day students. The first three stages mentioned do not follow a strict chronological order in the individual subject. The same student exhibits different levels of conceptualisation of negative numbers, depending on the task. For example, he may interpret 5-(-3) as 5-3, where the negative -3 is not ‘recognised’, while on the other hand, he ‘accepts’ the negative solution for some word problems, and rejects it in others. In the reification theory of Sfard and Linchevski (1994), the first four stages mentioned above could be described as operational, and the last as structural. The structural stage for negative numbers is usually not reached by 12-13 years old students. From the above work questions such as the following arose: with respect to equations and problems, what is the numerical domain that secondary school students ascribe to the constituent parts of an equation during the process of

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solution? Which numerical domain is accepted for the solution? What is the relationship between the numerical domain assigned to an equation and the type of language associated with the equation? Which methods or strategies obstruct or facilitate evolution towards the notion of number? These questions point to the need for research whose central concern is the study of the interrelationships between the processes of acquisition and use of algebraic language the methods for solving word problems and linear equations and the status of the negative number³ in word problems and linear equations. The general methodology of the project considers the interaction of these three components on two levels of analysis, the historical-critical level (evolution of meaning) and the didactic level (teaching-learning-cognition). With regard to the first of these, we found that the properties of subtraction, the laws of signs and certain elements necessary for operativity with negative numbers, appeared in remote historical times in the context of the solution of algebraic equations. The opposing concepts of gain and loss, property and debt, future and past, sale and purchase, are adequate interpretations for positive and negative. A crucial step towards acceptance of the negative number was to admit negative solutions to equations. The main difficulty facing medieval mathematicians in the solution of concrete problems was precisely the interpretation of negative solutions. The historical analysis carried out in this work allowed us to conclude that problems posed in old texts contributed to extend the numerical domain of natural numbers to that of the integers in the resolution of algebraic equations. In the following section we analyse three historical problems, one from the Chinese text Nine Chapters of Mathematical Art (ca. A.D. 250) one from the Hindu text Bijaganita (XIIth century) by Bhãskara and the third one from the French Triparty en la Science des Nombres (1484) by Nicolas Chuquet. Together with the historical analysis we also exhibit an analysis of the ways in which present-day 12-13 pupils approach the negative objects inherent in these problems. A%*+( &K"9D2/

The Fiu Zhang Suanshu (Nine Chapter of the Mathematical Art), is one of the earliest mathematical texts in China. Let us examine the eighth chapter entitled A"#$ E,/#$: Just like the other chapters in the text, the present version of the Fang Cheng chapter contains a number of problems together with their respective solutions. Firstly, we find the use of negative numbers, showing that the ancient Chinese had a clear concept of them and were able to apply negative numbers in mathematical problems as we would do nowadays. Secondly, the Fang Cheng chapter shows the formulations and solution of simultaneous linear equations of up to five unknowns. Thirdly, the Fang Cheng chapter introduces the methods of solving equations by tabulating the coefficients of unknowns and the absolute term in the forms of a

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matrix on the counting board, thereby facilitating the elimination of the unknowns, one by one. It must be emphasised that ancient civilisation had no ready made sets of notations. Conceptualisations were in a verbalised form, though the Chinese took a forward step when they used rod numerals to convert concepts onto the counting board. There are only two methods in this chapter. The first, called A"#$ E,/#$ or calculation by tabulation, is about solving a set of equations. The second method, called the positive-negative rules (zheng fu shu), comprises rules for the subtraction and addition of positive and negative numbers. The term Fang Cheng is defined as the arrangement of a series of things in columns for the purpose of mutual verification. The number of columns to be set up is determined by the number of things involved. In modem notation each column has two sections; the top section consists of the quantities aij (i, j = 1,2, ... n) of the various things while the bottom one shows the absolute terms bi (i = 1,2, ... n). Such an arrangement on the counting board can be shows as follows :

The whole process of operation is done on the counting board using the rod numerals to represent the various quantities.4 The unique place-value feature of this method of computation renders the use of symbols unnecessary. In each column of things on the counting board, the space between aij and bi, has the implicit function of an equal sign. The former matrix arrangement is transformed in such a way that all numbers in the upper side of the main diagonal are equal to zero (only columns are operated on). This transformed matrix corresponds to a diagonal set of equations, from which all the unknowns are successively determined. One can see that this method is essentially the usual method in present day algebra. Since the process of the Fang Cheng solution is the successive elimination of numbers through mutual subtraction of columns, there could be cases when a number to be subtracted from one column is smaller than the corresponding one in the other column. The opposite result obtained has to be indicated and certain rules have also to be established in order to continue the eliminating process. This gives

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rise to the creation of names: the term fu to indicate the resulting opposite amount to the term zheng for the normal difference. The concept of zheng and fu seems to have evolved from such ideas as ‘gain and ‘loss’ as clearly shown in Problem 8 which reads: ‘By selling 2 cows and 5 goats to buy 13 pigs, there is a surplus of 1000 cash. The money obtained from selling 3 cows and 3 pigs is just enough to buy 9 goats. By selling 6 goats and 8 pigs to buy 5 cows, there is a deficit of 600 cash. What is the price of a cow, a goat and a pig?’ The text considers the selling price zheng because of the money received and the buying price fu because of the money spent. The surplus amount is considered zheng and the deficit fu. These terms are merely names to indicate the nature of numbers. For the purpose of computation, numbers described by these terms have to be transcribed into a concrete form. There are two ways of doing this with rod numerals. If different coloured rods are used, then red ones represent zheng and black ones represent fu. Alternatively, if the rods are of one colour only, the fu numeral is indicated by an extra rod placed diagonally across its last non-zero digit. It is explained in the text that when a number is said to be negative, it does not necessarily mean that there is a deficit. Similarly, a positive number does not necessarily imply that there is gain. Therefore, even though there are red and black numerals in each column, a change in their colour resulting from the operations will not jeopardies the calculation. Negative numbers appeared as a c l a s s of numbers in the mathematical sense that is familiar to us today. The concept of positive and negative, which initially evolved from opposing entities such as ‘gain and loss’, ‘add and subtract’ and ‘sell and buy’, is now detached from linguistic associations. Its development has resulted in negative numbers being regarded as one group of numbers with properties which are connected with the other group of ‘normal’ or positive numbers. These properties are defined by positive-negative rules which correspond to the modern ones. Problem 8 involves selling and buying which equate to the concept of positive and negative respectively. The corresponding set of equations in tabulated form becomes: (3rd equation)

(2nd equation)

(1st equation)

(cows)

-5

3

2

(goats)

6

-9

5

(pigs)

8

3

-13

-600

0

1000

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When they had set up the equations in this form, they then operated the columns (in a way similar to the matrix reduction method for solving equations). They aim to make the numbers in the top right triangle zero. As can be seen, the Fiu Zhang has provided substantial evidence that, by the first century, the Chinese not only accepted the validity of negative numbers but understood their relationships with positive ones and were able to formulate rules and to compute with them. The historical analysis aforementioned suggested the use of what is called the ‘Chinese Model’ in the didactic realm. The 12-13 years students selected for this empirical study were familiar with the model of the number line and with the syntactical rules for operating on integers. These students showed conflict in the cases a-(-b) and -a-(-b) with a and b natural numbers. It was then decided to use a model which would give ‘concrete’ existence to negative numbers as the Chinese mathematicians did (they used black rods for negative numerals). The operativity employed in this teaching model corresponds to that of the Chinese mathematicians, that is, positive numbers are opposite to negative ones. The central concept arises, the sum of opposites is zero, which gives foundation to all the operations carried out within the model. The Chinese Model is based on 1) the counting of positive numbers extended to negative numbers; 2) an alternative representation of the minuend required in some cases in order to carry out the operation of taking away. Then, the addition of zeros is employed, as needed in each case. The students were presented with a diagrammatic version of the model. In the world of paper and pencils, positive numbers were white balls, and negative numbers, black balls. Zero was represented by the simultaneous presence of a white and a black ball. Operativity was carried out in the additive domain. For example, the addition 3 + (2) is presented; the numbers are described in the model

. They join together, provoking the

formation of zeros o . The result is a white ball which represents the number 1. Analogously, for the subtraction 2-3. The numbers are described in the model. Three cannot be subtracted from 2. A zero is added to the number 2. The representation of 2+ is obtained. Then the subtraction can be carried out, crossing out, A black ball is obtained, the representation of - 1. Some difficulties manifested by students when using the Chinese Model, are shown in what follows: The expression 2 - (-3) is represented as and zeros are formed . Instead of executing the action of taking away, the addition of opposites is effected.

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The diagrammatic representation of numbers in the model, is not adequately interpreted. For instance, a student says:

He does not identify the whole array as +1, because he fails to perceive opposites. On the other hand, some students identify opposites, but instead of cancelling, they try to give zero its positional value: is interpreted as 10. The following can be concluded from this analysis: competent users of the Chinese Model showed: differentiation of the actions of adding and subtracting negative numbers. recognition of the dual nature of zero, as a null element and formed by opposing elements (for example represents zero but also +l, - 1). distinction of negative numbers as relative numbers, and as isolated numbers (the students manifested five conceptual levels of negative numbers depending on the task, see page 128 of this chapter). It should be said that the main interest in modelling in this study is not its usefulness in teaching but as a resource which exhibits the different levels of acceptance of negative numbers by the students and allows them to develop different meanings for integer.5 ;/B)#' &K"9D2/

In the Hindu text Bijaganita (§140 in Colebrooke’s translation) Bhãskara treated equations of the second degree. He noted that, in some problems it may be that only one solution is acceptable even ifboth are positive. For example: ‘The fifth part of the troop less three, squared, had gone to a cave; and one monkey was in sight, having climbed on a branch. Say how many they were’. Bhãskara remarks: ‘Here a two-fold value is found, 50 and 5. But the second is in this case not to be taken: for it is incongruous. People do not approve a negative absolute number’. (The commentator of the text explains that the second value is five, its fifth part is one and can not be subtracted from three). The Hindu mathematicians used first letters of words for symbolising algebraic terms. They distinguished negative quantities by putting a point over the number and did not have a subtraction sign nor addition, multiplication or equal signs. The equation solving the problem above in both, Hindu’s notation and modern notation, is written as

I34

A. GALLARDO

where ya (the first letters from the word yávat-távat) is x, ya v represents x², ru (from the word rúpa) is used for the independent term and Va

is

Va

This problem was presented to present day 12-13 year old students. The lack of equilibrium between the semantics and syntax of algebraic language during the process of the solution’s validity was observed. The following is a fragment of an interview after a student had solved correctly the Bhãskara’s problem. I: There are two solutions. What does it mean? Do these express how many monkeys exist?How do you interpret this? S: Could it be possible for 50 and 5 to exist? 1: But, could both answers be correct? S: Yes, they could be both. I Is there anything that might hinder it? S: Oh, sure! -2 monkeys can’t ever exist. I: -2 monkey can’t ever exist! Where did 0)4 get that answer of -2 monkeys? (Then the student explains how he obtained -2 by substituting 5 in the equation +1

=

x

S: Because, 5 divided by 5 is equal to 1, minus three, is equal to minus 2. Thus, there couldn’t be -2 monkeys in a troops.

The student attributes total validity to algebraic methods and the process of verification. However the context prevents the acceptance of one of the two positive solutions. This happens during the process of verification where the subject cannot abandon the meaning of the symbol (monkeys) and give sense to the process. C,%*' &K"9D2/

In the Appendix of the text Triparty en la Science des Nombres, Chuquet exhibits problems in which a negative solution is accepted and this is interpreted according

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to the context of the problem. Chuquet introduced a syncopated language with a general notation very similar to modern symbolisation. He writes the numbers with a 1 zero exponent. for example 12 as the linear termx as 1 ,the square term x²as 1² and so on. He abandoned any geometric referent that was associated with x1 x ² in ancient times. Also he used the symbols p for the addition sign and m for the subtraction sign. There is no equal sign in equations. The following is an example of one of Chuquet’s problems: 5 9/*B,"#( ,"+ 3)4$,( Xa D%/B/+ )7 B2)(, "( (,/ D*%B/ )7 X]W B*)>#+ SdB4+T1 )#/ =%#' )7 >,%B, B)+(+ XX B*)>#+ "D%/B/ "#' (,/ )(,/* X[ B*)>#+: m/ 94+( '/(/*9%#/ (,/ */+D/B(%./ b4"#(%(%/+ )7 B2)(, D4*B,"+/': Chuquet’s solution was very similar to the following in modern notation.

C"=/ x = K X "+ (,/ 4#=#)>#: C,4+ K V @v Xa I K1 "#' (,/ +/B)#' /b4"(%)# 3/B)9/+ XXK + X[ (XaIKTv X]W1 >,/#B/

Having obtained 17 1/2 for one unknown he said: gJ)> +43(*"B( Xc XnV 7*)9 XaY (,/*/ */9"%# IV XnV D%/B/+ 4( (,/ D*%B/ )7 X[ E*)>#+ " D%/B/e: After verifying that the second equation is satisfied, Chuquet remarks that such problems are considered impossible. In that case, the impossibility (i.e. the occurrence of the negative result) is due, he observes, to the fact that 0 (crowns) does not fall between 11 and 13 crowns, the given prices. Chuquet proposes the following interpretation. ‘The merchant bought 171 pieces at 11 crowns per piece with cash, thus paying 192% crown. He then took 2% pieces at 13 crown per piece on credit to, the amount of 32% crowns. Thus he has a debt of 32 ½ crowns, the subtraction(!) of which from 192 ½ gives 160. Following the same reasoning, Chuquet considers that the 2% pieces bought on credit must be subtracted from the 17 ½ pieces purchased, and that the merchant has only 15 pieces which are properly his’. This problem was posed to 20 students of the research study being discussed in this section. Following are the methods used by students when solving Chuquet’s problem.

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5*%(,9/(%B !/(,)' (used by 15 students). The students looks for multiples of 11 and 13 that add up to 160. When the students do not find the multiples needed to solve the problem. that is XX x XX t X[ x [ v X]W1 they use an additional interpretation to explain their results, for example, Student 1. He writes 66 + 91 = 157, and says: ‘lie bought 6 pieces costing 11 coins and he had 3 coins left over’. Student 2. He writes 154 + 0 = 167, and explains ‘he bought 14 pieces costing 11 coins each and none costing 13 coins’. Student 3. He writes 154 + 13 = 167. and says ‘he owned 7 coins’. 5''%(%./ !/(,)' (used by one student). The problem of the purchase of goods is modified such that the figures arc smaller in order to facilitate solution. The equations which model the problem in this case are: K t 0 v [ Y VK t [0 v sW: The student assumes that each one of the prices of cloth has a price different from that established in the statement of the problem in order to adjust the total price. He writes, X K V t X K [ v a: thus, sW I a v [a: He then says, g (,/ +"2/+94# 3)4$,( [ D%/B/+P X B)+(%#$ V B)%#+1 "#)(,/* B)+(%#$ [ B)%#+ "#' " (,%*' B)+(%#$ [a B)%#+e:

;,"*%#$ !/(,)': This is also found in the modified version ( xt y v [ Y Vxt[yvsW ) : A student divides the total price, sW1 by two. The result of the division, VW, is used with the other data of the problem 2, 3, and he formulates the sums: Xl t V v VWY F c t [ v VW: His answer %+ g,/ 3)4$,( Xl D%/B/+ >)*(,V B)%#+ "#' F c D%/B/+ >)*(, [ B)%#+ /"B,e: It is important to point out that. contrary to what might be expected, the modified version of the statement (with small numbers) renders the problem impossible for many students. The conflict is accentuated since the solution is sought in the positive domain and the lack of adjustment between the data of the problem is more notorious than in the previous version ( x t y v XaY XXxtX[yvX]W ) where the magnitude of the numbers tends to hide the conflict. This obstacle disappears when it is suggested to the student that he uses algebra to solve the problem. 52$/3*"%B !/(,)' (used by two students). Spontaneous formulation of a system of equations to solve the problem. Let us now look at the case of a student who, by using the process of substitution of the solution in a system of equations, managed to solve the problem which at first he had thought impossible. The student formulates the equations XXx t X[y v X]WY x t y v Xa: He obtains the solution; x v Xc:a1 The following dialogue then ensued: S: Totally impossible I: And now, how are you going to find