Learning In and Out of School

M: Thirty-five.

CUSTOMER: I'd like ten. How much is that?

M: [Pause] Three will be one hundred and five; with three more, that will be two hundred and ten. [Pause] I need four more. That is ... [Pause] three hundred and fifteen...I think it is three hundred and fifty

As you can see, M correctly computes that 10 times 35 is 350. However, he doesn't use the procedure taught in Brazilian schools of simply placing a zero to the right of any number that is being multiplied by 10. Instead, he converts multiplication into repeated addition by threes 105 + 105 + 105 + 35. This is an example of an invented strategy used on an informal test of arithmetic.

M has had some schooling and currently is in the third grade. Suppose we visit him in school one day, give him a pencil and paper, and dictate some arithmetic problems and words problems to him. This is a formal test of arithmetic. For example, for the problem 35 X 4 = ________, M writes the answer "200." He explains his answer as follows:

Four times five is twenty, carry the two; two plus three is five, times four is twenty

As you can see, M is trying to use the school-taught procedure but because it is meaningless to him, he tends to makes some errors in applying it

This example comes from a study by Nunes, Schliemann, and Carraher (1993) in which they compare the performance of five children—all street vendors between ages 9 and 15—on formal and informal tests of arithmetic.The street vendors were nearly errorless in computing answers to arithmetic problems in the street but performed much more poorly when equivalent problems were presented in school-like form. In short, children who are "capable of solving a computational problem in the natural situation" often "fail to solve the same problem when it is taken out of context" (Nunes et al., 1993, p. 23).

What can we conclude from studies of Brazilian street vendors' These results demonstrate that "daily problem solving may be accomplished by routines different from those taught in schools" (Nunes et al., 1993, p. 26). In spite of the fact that the children in this study had received formal instruction in arithmetic computational procedures, they invented their own procedures to solve computational problems in the context of their roles as street vendors. Although they had difficulty in correctly applying school-taught procedures in a formal school-like context, they were highly successful in applying their own invented procedures in an informal everyday context These findings "raise doubts about the pedagogical practice of teaching children how to solve mathematical operations simply with numbers" (p. 25) and point to the role of cultural context in learning. In short, there is some support for students' claims that they do not use school-taught math outside of school.

This line of research has important implications for teaching that is intended to promote transfer. The major finding is that students often fail to transfer what they learned in school to problems outside of the school setting and often fail to transfer what they learned outside of school to solving problems in school. Thus, these findings indicate a need to teach in ways that promote transfer—that is, ways that help students be able to use what they learned in school when they are confronted with problems outside of school.

This example helps distinguish two views of the generality of learning. According to the classic view of learning, students abstract a general procedure from instruction and apply it across a wide variety of problems. Such a view would predict that people would use a school-taught procedure to solve the coconut problem. In contrast, the situated view of learning is that students acquire a specific procedure based on the context in which it was encountered and are able to use it mainly within that context. The coconut example supports a theory of situated learning—the idea that learning is shaped by and depends on the situation in which it takes place, including the social and cultural context of learning. An important challenge for educators is to create the kinds of social contexts that foster meaningful learning—that is, being able to use what is learned to solve new problems in new situations.