Mathematics In and Out of Context

By Benjamin M. on Friday, 4 August , 2006 - 11:47 pm

A contextual mathematical study was conducted by Nunes in Recife, Brazil.  The study focused on the children of street vendors who also assisted their parents in the day-to-day running of the business. The children were ages 8 and up. Children of certain socio-economic backgrounds may feel compelled to start their own business selling miscellaneous food items. Doing so requires mathematical skills including addition, subtraction, multiplication, and occasionally division. Children who run their own business also work with the concept profit and loss.

Of this study, Smith, Cowie and Blades (1998) extract the following:

“In the study there were four boys and one girl, age range 9-1, with a mean age of 11.2. All were from poor backgrounds. They were recruited to the study from street-corner stalls where they were working with their parents or alone. Test items were posed by the researchers posing as customers in the course of a normal transaction. At the end of the informal test, the children were asked to take part in a formal test a week later administered by the same researcher. There were 99 questions in the formal test and 63 questions in the informal test. The order of testing was the same for all participants.

The informal test
This was carried out in Portuguese in the naturalistic setting of the street-corner market. A researcher, posing as a customer, asked the children successive questions about potential purchases. The responses were written down by another researcher. After receiving the answer, the researcher asked the child how they had solved the problem. Here is an example of an informal test taken by M. a 12-year-old vendor:

Researcher: How much is one coconut?
M.: 35.
Researcher: 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… I think it is 350.

M. has solved the problem the following way:
(a) 35 x 10
(b) 35 X 3 (a sum which he probably already knew)
(c) 105 + 105
(d) 210 + 105
(e) 315 + 35
(f) 3 + 3 + 3 + 1

Even though he had been taught in school to multiply any number by 10 you simply add a zero to the right of that number, M. used a different problem-solving routine.

The formal test
After the test in a naturalistic setting, participants were invited to take part in the second part of the study. This took place on the street corner or at the child’s home. The items for the formal test were devised on the basis of the problems which the child had successfully solved in the naturalistic context. These test items were presented as 38 mathematical problems dictated to the child (e.g. 105 + 105) and 61 word problems (e.g. Mary bought x bananas; each banana costs y; how much did she pay altogether). In either case, the child solved problems involving the same numbers as those which were used in the informal test. The children were given paper and pencil, and were encouraged to use them if they wished. When the problems were solved mentally, the child was still asked to write down the answer. Only one of the children refused to do this on the ground that he did not know how to write.” (P. 449-450)

Table 1 illustrates how the problems, which arose in the context of the street market, were much more easily solved than those that were formally imposed in the second test.

Table 1 Results of the tests expressed in terms of percentage of correct items divided by 10

Child’s Initial Test taken informally Test taken formally
M 10 2.5
P 8.9 3.7
P’ 10 5.0
MD 10 1.0
S 10 8.3

Smith, Cowie and Blades further observe:

“In the informal test, 98.2 per cent of the 63 problems were correctly solved. By contrast, in the informal test, word problems (which provided some context) were correctly answered in 73.7 per cent of cases; mathematical problems with no context were solved in only 36.8 per cent of cases. The frequency of correct answers for each child was converted into scores from 1 to 10 reflecting the percentage of correct answers. A two-way analysis of variance of score ranks compared the scores of each participant in the three types of testing situation. The scores differ significantly across conditions (X2 = 6.4. p = 0.039). Mann-Whitney U’s were calculated. The children performed better on the informal test than on the formal test (U = 0. p <0.05).” (P. 450)

Process versus procedure

One explanation for this different between the two tests was that the errors were due to the modifications made between the informal test and formal test. Researchers considered this explanation and tested it by segregating formal test problems that had been modified from those that were merely a reproduction of the informal test. There was no sizeable difference in the participants’ ability to solve one type of problem over another.

Another hypothesis was that participants were still at the concrete thinking stage, in other words: they succeeded in the informal test because the products were concretely there for them to see. Nunes and her colleagues dismissed this explanation because the mere presence of food does not facilitate mathematical calculations.

A third possibility backed by a qualitative analysis of the interview procedures put forward that the children used different routines in each of the two tests. Informally, the children calculated using ‘convenient groups’; in the formal test they were using the school-taught procedures and formulas. There are several conclusive instances of this third explanation. Here is how the 12-year-old M proceeded for the same problem in the informal and formal tests
Informal Test
Researcher: I’m going to take four coconuts. How much is that?
M.: Three will be 105, plus 30, that’s 135… one coconut is 35… that is 140!

Formal test
M. (while attempting to solve 35 x 4): 4 times 5 is 20, carry the 2; 2 plus 3 is 5, times 4 is 20.
Written answer: 200.

The children were far more successful when dealing with familiar quantities and mental calculations. They multiplied by repeating additions and at times grouping quantities. In the formal context the students used procedures learnt in school with little understanding. The children were able to double-check their answers in the informal test because they were using a procedure that worked for them. In the formal context, the children parroted procedures and as a result, none showed enough comprehension to be able to check their answers in the formal tests.

Nunes and her colleagues conclude that the thought process that occurs from daily ‘common sense’ can be at a superior level than context-less thinking. The researchers raise concerns about introducing out-of-context problems before they are encountered in real life situations. Many a times, the school-learnt routines acted as interference between the child and the solution, children were oblivious to even absurd answers.

Nunes, Carraher and Schliemann believe that teachers have a role to play in helping learners develop their strategies. Mathematics taught in school should act as a driving force that works with the learner and builds on her existing strategies where possible. To do so, it must be taken from everyday situations and appeal to the children’s common sense. The study has shown that children can develop their own mathematical routines, which can be very different from those learnt in school.

In short, we can observe that schools try to impose procedures which are meaningless for the learner while she already has valid processes to draw from. These processes are not only ignored but discredited. Teachers would have a lot to gain from understanding the processes already in place in the learner so to construct upon it.

References:
Smith, P., Cowie, H., and Blades M. (1998), Understanding Children’s Development. Oxford: Blackwell Publishers
Nunes, T., Carraher, D. W., and Schliemann A. D. (1985) British Journal of Developmental Psychology, 3, 21-9

Category: Research

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