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New Directions lor Child Development, no. San Francisco: Fall Later, Piaget and Garcia proposed a cognitive explanation for the horizontal decalage, according to which development involved a progressive differentiation of the logico- mathematical structures from their contents; some contents, being more complex, are differentiated from the underlying structures later than the simpler ones. However, this explanation leaves no room for cultural practices and social situations to play a role in cognitive tasks, and ques- tions related to transfer across social situations remain unanswered.

There is considerable evidence of within-individual differences. Carraher, Carraher, and Schliemann , for example, observed that children who worked as street vendors were quite capable of solving arithmetic problems in the streets but appeared inept at solving problems zyx involving the same arithmetic operations in a school-like setting.

In different social situations, the same children show radically different per- formance in solving problems that relate to the same domain and pre- sumably call into play the same logico-mathematical structures. Lave, Murtaugh, and de la Rocha have also found large within-individ- ual differences among adults solving problems across situations: How can one understand substantial within-subject variation across social situations for the same type of prob- lems?

In other words, how is it possible that people who know how to solve a problem in one situation do not know how to solve the same problem in another situation?

In this chapter we will explore the relationship between concepts and the circumstances of learning in an attempt to understand questions related to within-individual variation. Vergnaud's framework will be used for analyzing how concepts relate across situations. After present- ing Vergnaud's basic ideas, we will use them to analyze some of our previous results on mathematical concepts in and out of school, discuss- ing their similarities and differences.

The first set of data refers to the solution of arithmetical problems. The second set concerns the solution of problems involving proportional reasoning. According to Vergnaud, a concept necessarily entails a set of invar- iants, which constitute the properties defining the concept, a set of signi- fiers, which are a particular symbolic representation of the concept, and a set of situations, which give meaning to the concept.

Due to the cen- trality of these three terms for the present analysis, we will consider each of them more closely. In the case of mathematical concepts, invariants correspond to mathe- matical properties. People often behave as if they knew about these invariants in the course of solving addition problems. A concept does not apply to one situation only but to several situa- tions that give meaning to the concept.

This is a new and important idea that Vergnaud introduces into con- ceptual analysis. Psychologists often view the defining properties of a concept as central aspects of concepts and extension as merely as epi- phenomenon.

Using addition once more as an example, the importance of situations in defining a concept can be clarified. Young children about 6 years old may understand the basic properties of addition and use them in solving problems. However, the set of situations to which addition is applied by six-year-olds tends to be limited. A mathematical concept may be represented, for example, through graphs, equations, or natural language.

Any representation is always only one of the possible zyxw representations of the same concept. Different representations of a concept tend to capture, in a clear fashion, different aspects of the concept. These signifiers do not represent the distinctions between situations in which the respective operations are useful. The sign -, for example, is used in situations in which we take away some- thing, describe a debt, carry out a comparison, refer to temperatures below zero, and so forth.

If all we know about a problem is that it involves -5, we cannot know what situation it refers to; the mathematical representation does not allow us to identify the particular problem- situation referred to.

When a characterization of differences in situations is necessary, other symbolic representations have to be introduced-like language. We will review data from our work on mathematics learned in and out of school in order to examine whether the same invariants under- lie the concepts learned in either setting, whether the use of different types of symbolic representation can account for within-individual differences across situations and whether street and school concepts differ in their generalizability across contents.

The first set of data concerns arithmetic operations, and the second set deals with proportional reasoning. The children in the first study Carraher, Carraher, and Schliemann, were engaged in the informal sector of the economy, selling fruits, vegetables, or popcorn. They had experience with arithmetic problem solving in and out of school. In their work as street vendors, they calculate the total costs of downloads zyx for example, the cost of twelve lemons and two avocados and the change due to their customers.

In school they solve computation exercises and word problems. These two social situations-street vending and school- ing-play a role in determining what type of symbolic representation is used for communication. In Brazilian street markets, written procedures are rarely used for calculating change. The currency itself supports the process of calculation: In schools, zyxw by contrast, written calculation is required correct numerical answers without the proper written calculation tend to be disregarded by teachers.

Children who are street vendors thus learn about arithmetic oper- ations under two different circumstances. Do they construct different in- variants for their work in and out of school? Should the difference in signifiers-written versus oral-affect their performance, or are the dif- ferences between oral and written modes irrelevant to how mathematical knowledge is used in problem solving?

In our first study Carraher, Carraher, and Schliemann, , we investigated the arithmetic problem-solving ability of five youngsters aged nine to fifteen years, with levels of schooling ranging from first to 75 zy zyx eighth grade in three conditions. In the streets, the youngsters were given problems in the course of a commercial transaction between the vendor and the experimenter-as-customer. The experimenter posed prob- lems about actual or possible downloads.

The problems of the street situation served as a basis for generating word problems and computation exercises, which were later presented to the same subjects in a school-like fashion. Striking within-subject variation in accuracy appeared across condi- tions: This compares to 74 percent when children worked on word problems and 37 percent correct answers on the computation exercises. While many differences across situations might account for the differences in performance for example, the exper- imenter-child relationship was different in the two situations , we noted qualitative differences in how the children represented the problems in the street and in the school-like situation.

Without exception, the chil- dren solved the problems in the street mentally, while in the school-like situation they often used paper and penal. We hypothesized that form of zyxwvu representation-oral versus written-had a strong impact on the differ- cal differences: Street Condition zyx ences in performance.

The following protocol illustrates one of the typi- zyxwvutsr Customer: What do I get back? Eighty, ninety, one hundred, four hundred and twenty. Formal Condition zyxwv Test item: The child writes plus 80 and obtains as the result. The child lowers the 0 and then apparently proceeds as follows: The result is Note that the child is applying steps from the multipli- cation algorithm to an addition problem. The experimenter was always an experimenter, not a customer, and the chil- zyxw dren were tested in their school.

Three interviewing conditions, counter- balanced to avoid order effects, were used for all children: The numbers in the arithmetic operations were the same across situations for different children, so that differences across condi- tions could not be attributed to differences in the values involved in the zyx problems. Paper and pencil were always available, but children could solve the problems in any way they wanted.

Significant differences in performance were again observed across examining conditions. First, experimental conditions were strongly related to the solution strategy. Children solved problems orally for over 80 percent of the simulated store problems and for 50 percent of the word problems but for less than 15 percent of the computation exercises.

Fur- ther, in each condition, oral calculations had higher success rates than written calculations. A repeated measures analysis of variance revealed significant differences in the percentage of correct responses as a function of testing condition: At first glance this would seem to suggest that children performed better under more concrete conditions.

However, when the oral and written procedures are separated within conditions, the differ- ences in success across conditions disappear. Thus the differences across conditions seem to be mediated by the type of solutions spontaneously adopted-oral or written. In other words, the same children solving prob- lems that required the same operations but using different representations showed very different performance.

The oral procedures used by the sixteen children were then analyzed more closely in order to understand why the symbols used in problem zyxw solving made such a difference.

Two general heuristics, decomposition and repeated groupings, were identified through this analysis. The following protocol exemplifies this heuristic. The child was solving a word-problem in which the subtraction - 35 was required. She said out loud: But it is Decomposition can be compared to the borrowing algorithm taught in Brazilian schools through a similar analysis into steps so that differ- ences that result from adding or subtracting orally or in writting can be identified and the invariants implicit in the two types of solution can be analyzed.

Children using the borrowing algorithm would write the number , write 35 underneath aligned from the right, and then go - through the following steps: The school algorithm can be rewritten as 1 is the same as - - f 10; 2 10 - 5 is 5; 3 9 from 3 is 6; 4 1 0 is 1; 5 read solution as The invariants implicit in the written and oral strategies can be stated as follows: This corre- sponds to the property of associativity-that is, the invariant underlying decomposition and addition or subtraction through written algorithms appears to be the same.

See Resnick, , for a similar analysis with an American child. Despite the use of the same invariant, differences can be pointed out that result from the use of oral or written signifiers. In the oral mode, the relative value of numbers is pronounced: In the written mode, the relative value is represented through rela- zyxwvu tive position: This difference in the signifiers is main- tained in the calculating procedures: This is clearly shown in the protocol below in which the same child solved the same problem in the oral and then in the written mode.

How did you do it so quickly? Two hundred, minus thirty, one seventy. Minus five, one sixty-five. Can you do it on paper? I used to know this.

Zero minus five, carry the one. Writes down 5 as the result for units. Cany the one writing down 7, apparently calculating 10 - 3. Carry the one. Two minus one. Writes down I ; the obtained result was When attempting the same problem in writing, the relative value was set aside and the wrong answer was obtained. The within-individual difference is all the more striking when the ease with which the child solves the computation in the oral mode is compared with the loss of meaning in the algorithmic procedure.

Many children appeared quite lacking in abil- ity when only their written attempts were considered, although they appeared quite at ease with numbers when only their oral calculations zyxwv were taken into account. Oral and written procedures also differ in the direction of calculation; the written algorithm is performed working from units to tens to hundreds, while the oral procedure follows the direction hundreds to tens to units. It involves repeated additions, in the case of mul- tiplication, and subtractions, in the case of division.

Repeated grouping, like the multiplication algorithm, relies on distributivity as an implicit zyxw invariant-as can be noted in this example of a child calculating 15 X 50 in the simulated-store condition: The child used in this multiplication the same groups we use in the written multiplication algorithm, namely, 5 and However, his oral multiplication was performed in the opposite order, using 10 and then 5 as factors. Further, the factor 10 preserved its relative value, 10 being pronounced as dez ten instead of urn one.

Thus multiplication and division replicate the previous observations with addition and subtrac- tion: However, it is possible that children behave as if they used associativity, commutativity, and distributivity but, in fact, they just have memorized procedures for calculating.

Hatano applied the distinc- tion between procedural and conceptual knowledge to the analysis of arithmetic operations, claiming that it is possible for subjects to learn how to carry out operators with the abacus without developing the cor- responding conceptual knowledge operations-displaying, thus, simply procedural knowledge.

Is it possible that the children who know oral mathematics merely 79 know routines for calculating? The flexibility of oral mathematics seems to be too great to fit the idea of simple routines; in fact, it is only by zyxw overlooking much variability of particular steps that general descriptions of oral heuristics are possible.

This flexibility contrasts strongly with the rigidity of the school-taught procedures, which may indeed be followed as routines without the corresponding comprehension. Cunha Hart , and Miranda have independently found that many children can carry out school-taught routines for adding and subtracting without understanding their mean- ing. We have so far examined the concepts of arithmetic operations. It has been argued Resnick, that children can learn these concepts outside school because the concepts are based on the additive composition prop- erty of numbers; more complex concepts, such as ratio and proportions, could not be understood in the absence of school instruction.

In the next section, two studies on the understanding of proportional relations devel- oped outside school are analyzed. The second study Schliemann and Carraher, ana- lyzes proportional reasoning among fishermen in the context of pricing and calculating net weight of seafood. Problem 1. When John was 13, Peter was John is now 23 years old.

How old is Peter? Problem 2. A wall drawn 6 cm long in a blueprint is 3 m long in reality. What is the real length of a wall, which is 10 cm long in the same blueprint? The contrast between the two problems shows that the invariant underlying solution in the first problem a constant difference differs from the invariant in the second a constant ratio. Data from two professions will be considered in this section in order to assess whether illiterate or semischooled adults who solve proportions problems in everyday life do so by constructing the appropriate invariants or by using procedural knowledge acquired outside school.

The studies explore the distinction between procedural and conceptual knowledge by testing for flexibility and transfer-which Hatano proposed as distinctive of conceptual knowledge. For exam- ple, people who usually calculate costs of downloads know the price of their merchandise per kilo and have to determine how much a greater number of kilos will cost. They can solve these problems by repeated addition or by multiplication.

In our experimental tasks, the subjects were told prices of larger amounts and asked to determine unit prices. Solving these problems would require division or subtraction-that is, the inverse of the usual procedures. In zyxwv the study about blueprints, which is described first, the transfer task zyxwv requires the subject to solve problems with new scales, for which familiar procedures could not work. In the second study, with fishermen, we inves- tigated the transfer from one content-the relationship between weight of unprocessed versus processed seafood.

Working with blueprints, foremen zyx learn about scales, which are a mathematical way of expressing the rela- tionship between the dimensions as drawn and the dimensions in reality. For example, a scale that is labeled 1 by 50 written as 1: Foremen in Brazil learn about scales on the job; they receive no training in school. In Recife the most common scales are 1: Foremen use their knowledge of scales in setting up guidelines to demarcate internal and external walls of buildings, making zyxw sure that length, width, and angles match specifications on the blueprint.

Although the life-size dimensions are often written on the blueprints next to the drawing of walls, it is not uncommon for foremen to have to calculate the width of a window or a hallway from the blueprint because that measurement was left out. The subjects were shown blueprints drawn to four different, unspecified scales and were 81 zy asked to determine the life-size measures of some walls in the blueprints, starting with three pieces of information: This task requires the foremen to invert their zyxw usual procedure in order to identify the scale used in the drawing.

Two problem-solving strategies accounted for approximately 94 per- zyxw cent of the responses given by foremen. It seemed as if our findings were Rorschach ink blots onto which readers projected their beliefs about social class, economic stratification, self- determination, and nature versus nurture. If so, findings from the field of Everyday Mathematics were likely to be used to promote ideologies rather than to better understand how mathematics is learned, taught and employed in and out of school.

As such, the work has been dealing with issues of importance to both theory and practice. It is fundamentally about trying to understand phenomena.

We spent many years trying to make sense of our initial observations and devising new studies, both in and out of school, to better understand the nature of mathematical reasoning. Much the contrary: we wanted to integrate our findings into long- standing traditions of learning, cognition, and epistemology.

We often framed problems in terms of oppositions everyday vs. But the presumption that the two sorts of mathematics whatever one wishes to label them are really one and the same, is equally problematic.

Admittedly, this is a compromise, but any pair of terms is going to raise issues. One needs to ask more nuanced questions to advance the research agenda. Vergnaud views concepts as consisting of three components: invariants, symbols, and situations. Platonism itself has long been held in high regard among professional mathematicians because it captures important aspects of the knowledge they most aspire to: timeless, ideal forms that cannot be directly apprehended through the perceptual apparatus.

The notion of anamnesis or reminiscence from the Meno dialogue strikes the modern researcher as downright mystical. On the whole, Platonism is silent regarding how people learn and teach mathematics. Carraher operations. We have made similar points for the invariants function and equation Carraher, Schliemann and Schwartz The same point could be made with regard to any mathematical object.

When a teacher draws a triangle on a sheet of paper, the object she is attempting to represent is not the drawn triangle itself but rather the idea of a triangle or a family of triangles. The drawn triangle is the signifier; the ideal triangle is the signified. Imagine a line drawn on a blackboard with numbers increasing in value from left to right. The chalk line is not the number line mathematicians talk about: it has a thickness and a fixed length, whereas the real number line5 has no thickness and it extends to infinity in both directions.

And, given any two points chosen on the real number line, there is always an infinite number of points and corresponding numbers in between. Even in elementary mathematics, it is important that students shift their attention toward ideas, relations and structures not available to direct perception.

Otherwise they run the risk of confusing that which is drawn, written, or uttered with the things they are meant to stand for, namely, mathematical objects and operations.

Vergnaud employs the term symbol in the broad sense of semiotics.

Symbols are signifiers that take on a variety of forms within and outside of mathematics. Symbol ic systems are structures that allow individual symbols to be composed, operated upon and interpreted within a set of conventions.

And symbolization is only part of although an important part of conceptualization. This is the most difficult component to understand, and Vergnaud has provided no more than a fleeting sketch. Nonetheless situations are critical to the present discussion. One often discusses situations as if they were places or occasions in which mathematical concepts are deployed. This is humorously evident in the early stages of learning, where irrelevant characteristics of situations are wedded to the concepts—for example, when young students believe that fractions are about pizzas or density is fundamentally about floating and sinking in water.

Number is introduced through counting things , rational numbers through the measurement of quantities. Early mathematics instruction often relies on modeling, with a curious twist: instead of simply applying previously learned mathematical methods to represent phenomena in the physical world, children acquire knowledge of mathematics through making sense of worldly phenomena. Beyond victims and noble savages 29 to extricate themselves from empirical observation, demonstration, and trial and error methods.

Mathematics must take on a life of its own, so to speak, and students need to develop an appreciation of validity independent of empirical corroboration. Likewise, they need to be able to derive new symbolic expressions from existing expressions by treating the written forms as syntactical objects, without having to imbue the forms with extra- mathematical meaning. How students make or fail to make such a transition is an important topic for research.

The street vendors were not operating directly on written symbols as pupils are taught to do in school. Maybe they were imagining actions involving currency and items downloadd. But such a system of representation would be worthless if vendors were not able to keep track of precise values or amounts—something very unlikely if their computations depended on mental images of physical objects.

When we looked closely at the intermediate values involved in their mental computations, it became clear that the mental algorithms were not the same ones taught in school. In addition and subtraction, for example, one performs column-wise computations proceeding from right to left. Our street vendors did not use such algorithms. They did compose and decompose amounts, but they did so in ways that did not quite match standard procedures taught in school.

And they often broke apart amounts opportunistically, in ways that made good use of the particular values at hand. He might then subtract 53 from , obtaining Next, he might subtract the 5 the remaining part of the subtrahend from , reaching an answer of Of course there were many ways that the subtrahend could have been broken up. But the way chosen allowed the problem-solver to pass, on the way to a solution, through the number Their mental representations of did not contain three digits.

They only needed to represent the hundreds, of which there was a single amount two ; there was no need to keep track of tens and units. It seemed that the number system of the street vendors was not a place value system6 at all! That is why the strings , , , , and represent unique values even though the digits are the same in each case.

Carraher So the mental arithmetic of street vendors involved somewhat different symbolic representations from those taught in school. We did not have direct access to their mental representations. But it began to appear reasonable that their number system was structured somewhat differently from the number system introduced in school.

I wondered how it was possible for two different symbolic systems to consistently produce the same, correct answers to arithmetic problems. This might have been obvious to those familiar with the history of mathematics. But it took some time for me to realize that each representational system would need to respect the same properties of arithmetic.

Here is where the notion of invariant proved useful. When talking about whole numbers and measures, addition does not depend on the order of the addends. This simple law,7 and others like it,8 cannot be violated without wreaking havoc on the results. But there are multiple ways to correctly enact arithmetic operations that respect the law. This line of interpretation suggests that the street vendors were comfortable with the commutative property of addition when doing oral mathematics.

The difficulties they exhibited with school algorithms seemed to be more tied to the symbolic procedures themselves or to other invariants. We should not assume that Everyday Mathematics is equal in scope, power, and efficiency to School Mathematics. Ask them whether they would prefer a coins worth 25 cents each or b 25 coins worth cents each, and you may find that, before they have determined the result of each case, they may suspect that one of the two options has a higher value.

If they do not have a method for performing multiplication, other than through repeated addition, they cannot be expected to be aware of the commutative property of multiplication. This brings us to the issue of the relative scope and power of Everyday Mathematics as compared to school mathematics. Beyond victims and noble savages 31 I fully agree with the authors on these points; my colleagues and I have made similar observations elsewhere Carraher et al.

Greiffenhagen and Sharrock are correct in noting that the threshold for success in supermarket mathematics was set considerably lower than for paper-and-pencil tests: declining to carry out a calculation in the supermarket was treated as appropriate for the setting but declining to provide an answer to an item on the paper and pencil arithmetic test was treated as an error. This makes comparisons of performance across situations misleading.

For these reasons alone it should not be regarded as an alternative to the mathematics found in textbooks. This is very different from elevating them to the goals of instruction.

However I would be disappointed if it is remembered only for its descriptive and proscriptive aspects, without recognizing the contributions to research, theory, and the cultural context of learning and thinking.

References Aleksandrov, A. A general view of mathematics S. Gould, T. Kirsh, Trans. Aleksandrov, A. Ball, D. Developing practice, developing practitioners: toward a practice-based theory of professional education. Darling-Hammond Eds. San Francisco: Jossey Bass. Bass, H.

On the field axioms and the commutative ring axioms. Email message to D. Carraher, Jan. Ann Arbor, MI. Carraher, D.

Oral and written mathematics.

Carraher Carraher, T. Carraher, T. Cadernos de Pesquisa, 42, 79— Mathematics in the streets and in schools.

British Journal of Developmental Psychology, 3 21 , 21— School failure: a social issue. Cadernos de Pesquisa, 45, 3—l9. Is everyday mathematics truly relevant to mathematics education?