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An analysis of everyday use of mathematics by working youngsters in commercial transactions in Recife, Brazil, revealed computational strategies different from those taught in schools. Performance on mathematical problems embedded in real-life contexts was superior to that on school-type word problems and context-free computational problems involving the same numbers and operations. Implications for education are examined.
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Mathematics in the streets and in schools
Terezinha Nunes Carraher; David William Carraher and Analucia Dias Schliemann
An analysis of everyday use of mathematics by working youngsters in commercial transactions in Recife,
Brazil, revealed computational strategies different from those taught in schools. Performance on mathematical
problems embedded in real-life contexts was superior to that on school-type word problems and context-free
computational problems involving the same numbers and operations. Implications for education are
There are reasons for thinking that there may be a difference between solving mathematical problems using
algorithms learned in school and solving them in familiar contexts out of school. Reed & Lave (1981) have
shown that people who have not been to school often solve such problems in different ways from people who
have. This certainly suggests that there are informal ways of doing mathematical calculations which have
little to do with the procedures taught in school.
Reed & Lave's study with Liberian adults showed differences between people who had and who had not
been to school. However, it is quite possible that the same differences between informal and school-based
routines could exist within people. In other words it might be the case that the same person could solve
problems sometimes in formal and at other times in informal ways. This seems particularly likely with
children who often have to do mathematical calculations in informal circumstances outside school at the same
time as their knowledge of the algorithms which they have to learn at school is imperfect and their use of
them ineffective.
We already know that children often obtain absurd results such as finding a remainder which is larger than
the minuend when they try to apply routines for computations which they learn at school (Carraher &
Schliemann, in press). There is also some evidence that informal procedures learned outside school are often
extremely effective. Gay & Cole (1976) for example showed that unschooled Kpelle traders estimated
quantities of rice far better than educated Americans managed to. So it seems quite possible that children
might have difficulty with routines learned at school and yet at the same time be able to solve the
mathematical problems for which these routines were devised in other more effective ways. One way to test
this idea is to look at children who have to make frequent and quite complex calculations outside school. The
children who sell things in street markets in Brazil form one such group (Carraher et af., 1982).
The cultural context
The study was conducted in Recife, a city of approximately 1.5 million people on the north-eastern coast of
Brazil. Like several other large Brazilian cities, Recife receives a very large number of migrant workers from
the rural areas who must adapt to a new way of living in a metropolitan region. In an anthropological study of
migrant workers in Sao Paulo, Brazil, Berlinck (I977) identified four pressing needs in this adaptation
process: finding a home, acquiring work papers, getting a job, and providing for immediate survival (whereas
in rural areas the family often obtains food through its own work). During the initial adaptation phase,
survival depends mostly upon resources brought by the migrants or received through begging. A large portion
of migrants later become unspecialized manual workers, either maintaining .a regular job or working in what
is known as the informal sector of the economy (Cavalcanti, 1978), The informal sector can be' characterized
as an unofficial part of the economy which consists of relatively unskilled jobs not regulated by government
organs thereby producing income not susceptible to taxation while at the same time not affording job security
or workers' rights such as health insurance. The income generated thereby is thus intermittent and variable.
The dimensions of a business enterprise in the informal sector are determined by the family's work capability.
Low educational and professional qualification levels are characteristic of the rather sizable population which
depends upon the informal sector. In Recife, approximately 30 per cent of the workforce is engaged in the
informal sector as its main activity and 18 per cent as a secondary activity (Cavalcanti, 1978). The
importance of such, sources of income for families in Brazil's lower socio-economic strata can be easily
understood by noting that the income of an unspecialized labourer's family is increased by 56 percent through
his wife's and children's activities in the informal sector in Sao Paulo; (Berlinck, 1977). In Fortaleza it
represents fully 60 per cent of the lower class
family's income (Cavalcanti& Duarte, 1980a).
Several types of occupationsdomestic work, street-vending, shoe-repairing and other types of small
repairs which are carried out without a fixed commercial addressare grouped as part of the informal sector of
the economy. The occupation considered in the present study-that of street-vendorsrepresents the principal
occupation of 10 per cent of the economically active population of Salvador (Cavalcanti & Duarte, 1980b)
and Fortaleza (Cavalcanti & Duarte, 1980a). Although no specific data regarding street-vendors were
obtained for Recife, data from Salvador and Fortaleza serve as close approximations since these cities are,
like Recife, State capitals from the same geographical region.
It is fairly common in Brazil for sons and daughters of street-vendors to help out their parents in their
businesses. From about the age of 8 or 9 the children will often enact some of the transactions for the parents
when they are busy with another customer or away on some errand. Pre-adolescents and teenagers may even
develop their own 'business', selling snack foods such as roasted peanuts, pop-corn, coconut milk or corn on
the cob. In Fortaleza and Salvador, where data are available, 2.2 and 1.4 per cent, respectively, of the
population actively engaged In the informal sector as street-vendors were aged 14 or less while 8.2 and 7.5
percent, respectively, were aged 15-19 years (Cavalcanti & Duarte, 1980a,b).
In their work these children and adolescents have to solve a large number of mathematical problems,
usually without recourse to paper and pencil. Problems may involve multiplication (one coconut cost x; four
coconuts, 4x), addition (4 coconuts and 12 lemons cost x + y), and subtraction (Cr$500i.e. 5OO cruzeiros
minus the purchase price will give the change due). Division is much less frequently used but appears in some
contexts in which the price is set with respect to a measuring unit (such as 1 kg) and the customer wants a
fraction of that unit: for example, when the particular item chosen weighs 1.2 kg. The use of tables listing
prices by number of items (one egg-12 cruzeiros; two eggs-24, etc.) is observed occasionally in natural
settings but was not observed among the children who took part in the study. Pencil and paper were also not
used by these children, although they may occasionally be used by adult vendors when adding long lists of
The children in this study were four boys and one girl aged 9-15 years with a mean age of 11.2 and ranging in
level of schooling from first to eighth grade, One of them had only one year of schooling; two had three years
of schooling; one, four years; and one, eight years. All were from very poor backgrounds. Four of the subjects
were attending school at the time and one had been out of school for two years, Four of these subjects had
received formal instruction on mathematical operations and word problems, The subject who attended first
grade and dropped out of school was unlikely to have learned multiplication and division in school since these
operations are usually initiated in second or third grade in public schools in Recife,
The children were found by the interviewers on street corners or at markets where they worked alone or with
their families. Interviewers chose subjects who seemed to be in the desired age range-school children or
young adolescents-obtaining information about their age and level of schooling along with information on the
prices of their merchandise, Test items in this situation were presented in the course of a normal sales
transaction in which the researcher posed as a customer. Purchases were sometimes carried out. In other cases
the 'customer' asked the vendor to perform calculations on possible purchases. At the end of the informal test,
the children were asked to take part in a formal test which was given on a separate occasion, no more than a
week later, by the same interviewer. Subjects answered a total of 99 questions on the formal test and 63
questions on the informal test. Since the items of the formal test were based upon questions of the informal
test, order of testing was fixed for all subjects.
In the present report the term 'class' is employed loosely, without a clear distinction from the expression
'socio-economic stratum'.
(I) The informal test, The informal test was carried out in Portuguese in the subject's natural working
situation, that is, at street corners or an open market. Testers posed to the subject successive questions about
potential or actual purchases and obtained verbal responses. Responses were either tape-recorded or written
down. along with comments, by an observer. After obtaining an answer for the item, testers questioned the
subject about his or her method for solving the problem.
The method can be described as a hybrid between the Piagetian clinical method and participant observation.
The interviewer was not merely an interviewer; he was also a customera questioning customer who wanted
the vendor to tell him how he or she performed their computations.
An example is presented below taken from the informal test with M., a coconut vendor aged 12, third
grader, where the interviewer is referred to as 'customer':
Customer: How much is one coconut?
M: 35.
Customer: I'd like ten. How much is that?
M. (Pause) Three will be 105; with three more, that will be 210. (Pause) I need four more. That is…
(pause) 315…1 think it is 350.
This problem can be mathematically represented in several ways: 35 x 10 is a good representation of the
question posed by the interviewer. The subject's answer is better represented by 105 + 105 + 105 + 35, which
implies that 35 x 10 was solved by the subject as (3 x 35) + (3 x 35) + (3 x 35) + 35. The subject can be said
to have solved the following subitems in the above situation:
(a) 35 x 10;
(b) 35 x 3 (which may have already been known);
(c) 105 + 105;
(d) 210 +105;
(e) 315 +35;
(f) 3 + 3 + 3 + 1.
When one represents in a formal mathematical fashion the problems which were solved by the subject, one is
in fact attempting to represent the subject's mathematical competence. M. proved to be competent in finding
out how much 35 x 10 is, even though he used a routine not taught in third grade, since in Brazil third-graders
learn to multiply any number by ten simply by placing a zero to the right of that number. Thus, we considered
that the subject solved the test item (35 x 10) and a whole series of sub-items (b to f) successfully in this
process. However, In the process of scoring, only one test item (35 x 10) was considered as having been
presented and. therefore, correctly solved.
(2) The formal test. After subjects were interviewed in the natural situation, they were asked to participate in
the formal part of the study and a second interview was scheduled at the same place or at the subject's house.
The items for the formal test were prepared for each subject on the basis of problems solved by him or her
during the informal test. Each problem solved in the informal test was mathematically represented according
to the subject's problem-solving routine.
From all the mathematical problems successful/y solved by each subject (regardless of whether they
constituted a test item or not), a sample was chosen for inclusion in the subject's formal test. This sample was
presented in the formal test either as a mathematical operation dictated to the subject (e.g. 105 + 105) or as a
word problem e.g, Mary bought x bananas; each banana cost y; how much did she pay altogether?). In either
case, each subject solved problems employing the same numbers involved in his or her own informal test.
Thus quantities used varied from one subject to the other.
Two variations were introduced in the formal test, according to methodological suggestions contained in
Reed & Lave (1981). First, some of the items presented in the formal test were the inverse of problems solved
in the informal test (e.g. 500 - 385 may be presented as 385 + 115 in the formal test). Second, some of the
(…) is used here to mark ascending intonation suggestive of the interruption, and not completion, of a statement.
items in the informal test used a decimal value which differed from the one used in the formal test (e.g. 40
cruzeiros may have appeared as 40 centavos or 35 may have been presented as 3500 in the formal test--the
principal Brazilian unit of currency is the cruzeiro; each cruzeiro is worth one hundred centavos).
In order to make the formal test situation more similar to the school selling, subjects were given paper and
pencil at the testing and were encouraged to use these. When problems were nonetheless solved without
recourse; to writing, subjects were asked to write down their answers. Only one subject refused to do so,
claiming that he did not know how to write. It will be recalled, however, that the school-type situation was not
represented solely by the introduction of pencil and paper but also by the very use of formal mathematical
problems without context and by word problems referring to imaginary situations.
In the formal test the children were given a total of 38 mathematical operations and 61 word problems.
Word problems were rather concrete and each involved only one mathematical operation.
Results and discussion
The analysis of the results from the informal test required an initial definition of what would be considered a
test item in that situation. While, in the formal test, items were defined prior to testing, in the informal test
problems were generated in the natural setting and items were identified a posteriori. In order to avoid a
biased increase in the number or; items solved in the informal test, the definition of an item was based upon
questions posed by the customer/tester. This probably constitutes a conservative estimate of the number of
problems solved, since subjects often solved a number of intermediary steps in the course of searching for the
solution to the question they had been asked. Thus the same defining criterion was applied in both testing
situations in the identification of items even though items were defined prior to testing in one case and after
testing in the other. In both testing situations, the subject's oral response was the one taken into account even
though in the formal test written responses were also available.
Context-embedded problems were much more easily solved than ones without a context. Table I shows that
98.2 per cent of the 63 problems presented in the informal test were correctly solved. In the formal test word
problems (which provide some descriptive context for the subject), the rate of correct responses was 73.7 per
cent, which should be contrasted with a 36.8 per cent rate of correct responses for mathematical operations
with no context.
Table 1. Results according to testing conditions
Formal Test
Informal test
Word Problems
Number of
Number of
Number of
"Each subject's score is the percentage of correct items divided by 10.
The frequency of correct answers for each subject was converted to scores from I to 10 reflecting the
percentage of correct responses. A Friedman two-way analysis of variance or score ranks compared the scores
of each subject in the three types of testing conditions. The scores differ significantly across conditions
r=6.4, P=0.039). Mann-Whitney Us were also calculated comparing the three types of testing situations.
Subjects performed significantly better on the informal test than on the formal test involving context-free
operations (U = 0, P < 0.05). The difference between the informal test and the word problems was not
significant (U = 6, P> 0.05).
It could be argued that errors observed in the formal test were related to the transformations that had been
performed upon the informal test problems in order to construct the formal test. An evaluation of this
hypothesis was obtained by separating items which had been changed either by inverting the operation or
changing the decimal point from items which remained identical to their informal test equivalents. The
percentage of correct responses in these two groups of items did not differ significantly; the rate of correct
responses in transformed items was slightly higher than that obtained for items identical to informal test
items. Thus the transformations performed upon informal test items in designing formal test items cannot
explain the discrepancy of performance in these situations. .
A second possible interpretation of these results is that the children interviewed in this study were 'concrete'
in their thinking and, thus, concrete situations would help them in the discovery of a solution. In the natural
situation, they solved problems about the sale of lemons, coconuts, etc., when the actual items in question
were physically present. However, the presence of concrete instances can be understood as a facilitating factor
if the instance somehow allows the problem-solver to abstract from the concrete example to a more general
situation. There is nothing in the nature of coconuts that makes it relatively easier to discover t4at three
coconuts (at Cr$ 35.00 each) cost Cr$ 105.00. The presence of the groceries does not simplify the arithmetic
of the problem. Moreover, computation in the natural situation of the informal test was in all cases carried out
mentally, without recourse to external memory aids for partial results or intermediary steps. One can hardly
argue that mental computation would be an ability characteristic of concrete thinkers.
The results seem to be in conflict with the implicit pedagogical assumption of mathematical educators
according to which children ought first to learn mathematical operations and only later to apply them to verbal
and real-life problems. Real-life and word problems may provide the 'daily human sense' (Donaldson, 1978)
which will guide children to find a correct solution intuitively without requiring an extra step-namely, the
translation of word problems into algebraic expressions, This interpretation is consistent with data obtained
by others in the area of logic, such as Wason & Shapiro (1971), Johnson-Laird et al. (1972) and Lunzer et al.
How is it possible that children capable of solving a computational problem in' the natural situation will fail
to solve the same problem when it is taken out of its context? In the present case, a qualitative analysis of the
protocols suggested that the problem-solving routines used may have been different in the two situations. In
the natural situations children tended to reason by using what can be termed a 'convenient group' while in the
formal test school-taught routines were more frequently, although not exclusively, observed. Five examples
are given below, which demonstrate the children's ability to deal with quantities and their lack of expertise in
manipulating symbols. The examples were chosen for representing clear explanations of the procedures used
in both settings. In each o_ the five examples below the performance described in the informal test contrasts
strongly with the same child's performance in the formal test when solving the same item.
(1) First example (M, 12 years)
Informal test
Customer: I'm going to take four coconuts. How much is that?
Child: Three wil1 be 105, plus 30, that's 135 . . . one coconut is 35… that is . . .
Formal test
Child resolves the item 35 x 4 explaining out loud:
4 times 5 is 20, carry the 2; 2 plus 3 is 5, times 4 is 20. Answer written: 200.
(2) Second example (MD, 9 years)
Informal test
Customer: OK, I'll take three coconuts (at the price of Cr$ 40.00 each). How much is
Child: (Without gestures, calculates out loud) 40, 80, 120.
Formal test
Child solves the item 40 x 3 and obtains 70. She then explains the procedure 'Lower the zero;
4 and 3 is 7’.
(3) Third example (MD, 9 years)
Informal test
Customer: I'll take 12 lemons (one lemon is Cr$ 5.00)
Child: 10, 20, 30, 40, 50, 60 (while separating out two lemons at a time).
Formal test;
Child has just solved the item 40 x 3. In solving 12 x 5 she proceeds by lowering first the 2, then
the 5 and the 1, obtaining 152. She explains this procedure to the (surprised) examiner when she
is finished.
(4) Fourth example (S, II years)
Informal test
Customer: What would I have to pay for six kilos? (of watermelon at Cr$ 50.00 per kg).
Child: [Without any appreciable pause] 300.
Customer: Let me see. How did you get that so fast?
Child: Counting one by one. Two kilos, 100. 200. 300.
Formal test
Test item: A fisherman caught 50 fish. The second one caught five times the amount of
fish the first fisherman had caught. How many fish did the lucky fisherman
Child: (Writes down 50 x 6 and 360 as the result; then answers) 36.
Examiner repeats the problems and child does the computation again, writing down 860 as
result. His oral response is 86.
Examiner: How did you calculate that?
Child: I did it like this. Six times six is 36. Then I put it there.
Examiner: Where did you put it? (Child had not written down the number to be carried.)
Child: (Points to the digit 5 in 50). That makes 86 [apparently adding 3 and 5 and
placing this sum in the result].
Examiner: How many did the first fisherman catch? Child: 50.
A final example follows, with suggested interpretations enclosed in parentheses.
(5) Fifth example
Informal test
Customer: I'll take two coconuts (at Cr$ 40.00 each. Pays with a Cr$ 500.00 bill). What
do I get back?
Child: (Before reaching for customer's change) 80, 90, 100. 420.
Formal test
Test item: 420 + 80.
The child writes 420 + 80 and claims that 130 is the result. [The procedure used was not
explained but it seems that the child applied a step of a multiplication routine to an addition
problem by successively adding 8 to 2 and then to 4, carrying the 1; that is, 8 + 2 = 10, carry the
one, 1 + 4 + 8 = 13. The zeros in 420 and 80 were not written. Reaction times were obtained
from tape recordings and the whole process took 53 seconds.]
Examiner: How did you do this one, 420 plus 80?
Child: Plus?
Examiner: Plus 8O.
Child: 100, 200.
Examiner: (After a 5 second pause, interrupts the child's response treating it as final)
Hum, OK.
Child: Wait a minute. That was wrong. 500. [The child had apparently added 80 and
20, obtaining one hundred, and then started adding the hundreds. The
experimenter interpreted 200 as the final answer after a brief pause but the
child completed the computation and gave the correct answer when solving
the addition problem by a manipulation-with-quantities approach.]
In the informal test, children rely upon mental calculations which are closely linked to the quantities that are
being dealt with. The preferred strategy for multiplication problems seems to consist in chaining successive
additions. In the first example, as the addition became more difficult, the subject decomposed a quantity into
tens and units-to add 35 to 105, M. first added 3O and later included 5 in the result.
In the formal test, where paper and pencil were used in all the above examples, the children try to follow,
without success, school-prescribed routines. Mistakes often occur as a result of confusing addition routines
with multiplication routines, as is clearly the case in examples (1) and (5). Moreover, in all the cases, there is
no evidence, once the numbers are written down, that the children try to relate the obtained results to the
problem at hand in order to assess the adequacy 6f their answers.
Summarizing briefly, the combination of the clinical method of questioning with participant observation
used in this project seemed particularly helpful when exploring mathematical thinking and thinking in daily
life. The results support the thesis proposed by Luria (1976) and by Donaldson (1978) that thinking sustained
by daily human sense can be-in the same subject-at a higher level than thinking out of context. They also raise
doubts about the pedagogical practice of teaching mathematical operations in a disembedded form before
applying them to word problems.
Our results are also in agreement with data reported by Lave et al. (1984), who showed that problem
solving in the supermarket was significantly superior to problem solving with paper and pencil. It appears that
daily problem solving may be accomplished by routines different from those taught in schools. In the present
study, daily problem solving tended to be accomplished by strategies involving the mental manipulation of
quantities while in the school-type situation the manipulation of symbols carried the burden of computation,
thereby making the operations 'in a very real sense divorced from reality' (see Reed & Lave, 1981, p. 442). In
many cases attempts to follow school-prescribed routines seemed in fact to interfere with problem solving
(see also Carraher & Schliemann, in press).
Are we to conclude that schools ought to allow children simply to develop their own computational
routines without trying to impose the conventional systems developed in the culture? We do not believe that
our results lead to this conclusion. Mental computation has limitations which can be overcome through
written computation. One is the inherent limitation placed on multiplying through successive chunking, that
is, on multiplying through repeated chunked additionsa procedure which becomes grossly inefficient when
large numbers are involved.
The sort of mathematics taught in schools has the potential to serve as an 'amplifier of thought processes', in
the sense in which Bruner (1972) has referred to both mathematics and logic. As such, we do not dispute
whether 'school maths' 'routines can offer richer and more powerful alternatives to maths routines which
emerge in non-school settings. The major question appears to centre on the proper pedagogical point of
departure, i.e. where to start. We suggest that educators should question the practice of treating mathematical
systems as formal subjects from the outset and should instead seek ways of introducing these systems in
contexts which allow them to be sustained by human daily sense.
The research conducted received support from the Conselho Nacional de Desenvolvimento Cientifico e
Tecnologlco, Brasilia, and from the British Councll.
The authors thank Peter Bryant for his helpful comments on the present report.
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Received 13 July 1983; revised version received 20 July 1984
Requests for reprints should be addressed to Dr T. N. Carraher. Universidade Federal de Pernambuco, Rua
Mendes Martins, 112, Várzea, 50 000 Recife PE, Brazil.
D. W. Carraher and A. D. Schliemann are also at the Universidade Federal de Pernambuco, Recife.
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... The social marking of knowledge in Brazil was also documented in the domain of numeracy. Nunes, Schliemann, and Carraher (1993) observed that young street vendors were quite capable of solving arithmetic problems when selling produce in street markets (an average of 98% success). However, when asked to solve sums in a test-like situation, instead of using oral calculation methods, which they used so successfully when calculating in the street market, most of the time the youngsters chose to use written algorithms, which they did not master and often led to error (an average of only 37% success). ...
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Nunes, T., & Roazzi, A. (2000). Education, social identity and occupational aspirations in Brazil: Reasons for (not) learning. Em Fiona E. Leach & A. W. Little (Eds.), Education, Cultures and economics: Dilemmas for development (Chapter 18 - pp. 327-346). New York: Routledge. /// eBook Published 2013. ISBN: 1135651779, 9781135651770 Doi: /// Abstract: The analysis of the intergenerational transmission of social status and its relationship to education has seen waves of research and theory with markedly different orientations. The early evidence of an association between performance in intelligence tests and social class (Terman & Merril 1937), coupled with a view of society as a meritocracy (Herrnstein 1971), made this intergenerational transmission unproblematic in the past. Schools could be treated as a black box (French 1990), where children learned according to their abilities and their motivation; once they left school, society placed them in jobs that were adequate for their educational achievement. Theory and research were then geared to determining how much of the observed differences in intelligence tests were a consequence of heredity and how much resulted from the environment. With the increased investments in compensatory education in the 1960s and the allegedly small return in changing the intergenerational transmission of social status and its educational component ("Compensatory education has been tried and it has failed" ; Jensen 1969, 2), some researchers became aware of the need for a paradigm shift. The role of schools in the intergenerational transmission of educational failure (Bourdieu and Passeron 1977) was then recognized and documented through sociological analyses of who succeeds and who fails. However, these analyses do not tell the whole story because there is no strict determinism in social and educational status across generations in democratic societies (Bourdieu 1974), and researchers turned once again to closer analyses of individuals, their backgrounds and their trajectories through school.
... Therefore, the process of interpreting media graphs is not just made up of formal mathematical knowledge. These approaches were then similar to studies that discussed the relationship between schooling and the use of out-of-school knowledge in everyday social situations (Nunes, Schliemann, & Carraher, 1993). ...
The following paper discusses the status of intelligence research in the Anthropocene. First, I discuss how the transformations we have experienced signal the need to more deeply consider the role of context in our thinking of intelligence. Next, I discuss the demographic and cultural changes that transformed the niche of human intelligence after the Industrial Revolution. Then, I comment on how, in the origin of intelligence research, the invention of the theory of general intelligence was marked by a lack of consideration of the role of context, notwithstanding that the British founders of the field were working in the midst of the great transformation provoked by the Industrial Revolution. Finally, I conclude by discussing how intelligence research should be conducted to address the challenges of the Anthropocene.KeywordsIntelligenceCultureAnthropocene
This chapter begins by describing what is unique about mathematics that has made it a central topic in the learning sciences. This research has historically been interdisciplinary, drawing on psychology, mathematics research and theory, and mathematics educators. It then describes two distinct approaches – the acquisitionist and the participationist. The acquisitionist approach considers learning to be what happens when an individual learner acquires mathematical knowledge. This part of the chapter reviews research on misconceptions and conceptual change that has been based in Piaget’s constructivist theories. The participationist approach views learning as originating in social interactions in diverse settings such as classrooms, homes and playgrounds, museums, and workplaces. This approach views learning as a collective sociocultural phenomenon, and uses methodologies such as interaction analysis and design-based research. This chapter concludes with a discussion of how teachers learn to teach mathematics.
Artan göç dalgalarıyla birlikte Türkiye’deki okulların kültürel ve etnik yapısı gün geçtikçe daha çeşitli hale gelmiş ve adil matematik öğrenme fırsatlarına odaklanma gerekliliği ortaya çıkmıştır. Bu çalışmanın amacı sınıf öğretmenlerinin ve ortaokul matematik öğretmenlerinin kültürel çeşitlilik gösteren öğrencilerin matematik öğrenme fırsatlarına yönelik bakış açılarını araştırmak ve matematiği erişilebilir kılmak için belirledikleri pedagojik stratejileri ortaya çıkarmaktır. Bu doğrultuda nitel bir anket formu hazırlanarak 83 öğretmene ulaşılmıştır. Yapılan analizlerin sonucunda, çalışmaya katılan öğretmenler kültürel çeşitlilik gösteren öğrencilerin matematik derslerine katılımının daha az ve matematik öğretirken en zorlanılan konunun ise dil farklılıkları ve sınıf içi iletişim problemi olduğunu ifade ederken farklı kültürlere sahip öğrencilerinden temel matematik bilgi ve becerileri öğrenmesini beklemektedirler. Kültürel olarak çeşitli olan sınıflarda matematiğe adil erişim için kullanılan pedagojik stratejiler arasında temel olarak görsel materyal ve resimlerin kullanma, seviyelerine göre farklılaştırma ve günlük yaşantılarından örnekler verme sayılabilir. Ancak katılımcıların dörtte biri matematik öğrenmek için Türkçe dilinin öğrenilmesi gerektiğini savunmuşlardır. Bulgular ışığında, kültürel ve dilsel olarak duyarlı matematik pedagojilerinin çok kültürlü sınıflarda görev yapan öğretmenler ile mesleki gelişim projeleriyle paylaşılması ve uygulamaya koyulması önerilebilir. Ayrıca matematik öğrenme fırsatlarını derinlemesine inceleyen nitel çalışmalara ihtiyaç vardır.
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Compreendendo a educação como um Direito Humano fundamental para a formação de cidadãos acredita-se que é imprescindível que todas as pessoas, independentemente de suas diferenças tenham acesso às escolas, espaços formais de educação, e aprendam os conteúdos previstos nas diretrizes vigentes, como prevê a educação inclusiva. A educação inclusiva tem o desafio de ensinar estudantes diferentes, com cultura, hábitos, costume e condições socioeconômicas, religião e ritmos e maneiras diferentes de aprender, decorrentes ou não de deficiência. Destacando a Matemática, como uma área de conhecimento prevista na educação básica e os desafios de trabalhar os conteúdos desta com a diversidade de estudantes encontradas nas escolas, o presente artigo resulta de um estudo exploratório descritivo de caráter qualitativo, que tem como objetivo geral descrever uma proposta de prática docente para ensino de números decimais na perspectiva do Desenho Universal para a Aprendizagem (DUA). O DUA reconhece a pluralidade humana, e que cada um se apropria do conhecimento de maneiras diferentes. E, tem como preceito tornar o currículo básico da educação acessível e aplicável a todos, estruturado em princípios que serviram de base para se organizar atividades para ensinar números decimais utilizando materiais concretos, aula e práticas condizentes com o interesse e realidade dos estudantes, de modo a estimular e garantir aprendizagem, planejando e estruturando a ação docente para que esta seja acessível para todos, contemplando a diversidades de estudantes. A elaboração de proposta de atividades de Matemática na perspectiva do DUA demonstra que para atuar na diversidade o professor deve focar no estudante, organizando atividades com recursos que estimulem o aprendizado, que sejam contextualizadas e possibilitem ao estudante ver aplicabilidade do que aprende.
Fifty Brazilian children between the ages of 7 and 13, all number conservers, were individually given seven addition and four subtraction exercises. The children were asked to write down and do each exercise and then explain how the answer was reached. The results showed that: (a) counting was the preferred procedure; (b) the use of school-taught algorithms was rather limited; (c) some children decomposed numbers into tens and units and then worked at both levels, combining results subsequently; and (d) children rarely referred to previously obtained results when doing related exercises. Mathematics educators could profit from a knowledge of the procedures used naturally by children and the specific difficulties related to natural and to school-prescribed routines.
Three experiments were designed to investigate the failure of intelligent adults to solve an apparently simple problem of formal reasoning devised by Wason. Both the mode of presentation and the type of material reduced the difficulty of the problem, while retaining its essential form. However, success on the original problem remained at a low level, even when subjects had attempted an easier version and had been given an explanation. These results enable one to reject a “strong” formulation of Piaget's theory of formal reasoning. A “weaker” formulation is suggested as a basis for further research.
This study is concerned with the effects of prior experience on a deceptive reasoning problem. In the first experiment the subjects (students) were presented with the problem after they had experienced its logical structure. This experience was, on the whole, ineffective in allowing subsequent insight to be gained into the problem. In the second experiment the problem was presented in “thematic” form to one group, and in abstract form to the other group. Ten out of 16 subjects solved it in the thematic group, as opposed to 2 out of 16 in the abstract group. Three hypotheses are proposed to account for this result.
An experiment was performed to determine whether the use of realistic materials would improve performance in a deceptive reasoning problem. The task involved selecting from a set of envelopes those which, if they were turned over, could violate a given rule. The rule concerned either a realistic relation (‘if a letter is sealed, then it has a 50 lire stamp on it’) or else an arbitrary relation between symbols (‘if a letter has an A on one side, then it has a 3 on the other side’). Twenty-two of the 24 subjects made at least one correct answer with the realistic material but only seven of them did so with the symbolic materials. The verbal formulation of the rule was also varied but yielded only a marginal interaction with the main variable. It is argued that the critical factor is the intrinsic connexion between items rather than their specific nature.
After working out the details of four arithmetic systems among traditional tribal tailors in West Africa, we present an analysis of arithmetic errors that makes possible predictions about the varied educational experiences of the tailors and about individual problem-solving characteristics. It is argued that arithmetic is a powerful domain for exploring the impact of education on cognitive skills because of the systematic properties of the domain and because coexisting arithmetic systems lead to contrasting problem-solving strategies within most cultures.