Academic journal article
By Bruno, Alicia; Martinon, Antonio; Velazquez, Fidela
Focus on Learning Problems in Mathematics , Vol. 26, No. 1
The research carried out on simple addition and subtraction word problems (solved by x + y = z or x - y = z) has been very extensive. Ample bibliographical details on this subject may be found in research surveys by Fuson (1992) and Verschaffel and De Corte (1996).
From experience, and the results of research, we know that each student has a varying degree of problem-solving success with different problems and also that different students have different levels of success in each problem. These facts are explained through different problem characteristics. Several classes of additive problems are well known: Combine, Change, Compare and Equalize (Carpenter and Moser, 1982; Riley et al., 1983). In this paper we are particularly interested in certain aspects of the problems that we now summarize.
In certain numerical situations two states are compared: a small state ("Juan has 2 pesetas") and a big state ("Pedro has 5 pesetas"). We can use the scheme s + d = b, where s and b are static situations and d is the difference. There are two ways in which the difference may be expressed. In Compare problems, the difference is expressed as "more than" ("Pedro has 3 pesetas more than Juan") or "less than" ("Juan has 3 pesetas less than Pedro"). In Equalize problems, the expression would be "how much" the small state must increase to equalize the big state ("If Juan earns 3 pesetas, then he has the same as Pedro") or "how much" the big state must decrease to equalize the small state ("If Pedro loses 3 pesetas, then he has the same as Juan").
In other situations we have a start state ("Before, Juan had 2 pesetas"), a variation ("then he earned 3 pesetas") and an end state ("Juan has now 5 pesetas"). These problems have the scheme s + v = e and are associated with dynamic situations. There are two types of expression for the variation: in Change problems, the variation is expressed in a simple way ("Juan has earned ..." or "Juan has lost ..."). In Change-Compare problems the variation is expressed as more than or less than, in a similar way to Compare problems ("Now, Juan has 3 pesetas more than he had before"). We don't know of any research study that covers Change-Compare problems. These classes of problems are shown in Table 1.
The above distinction between scheme and expression is not usual. To sum up, in an additive situation where three numbers are involved: a + b = c, the scheme refers to the numerical situation and the expression refers to the manner of saying (or writing) the variation and the difference.
Fuson and Willis (1986) noted that Compare and Equalize problems have different problem-solving difficulty levels: Compare problems are generally more difficult to solve than Equalize problems; the expression of the difference therefore influences the level of difficulty. In the present research, we show that the expression of the variation, in Change problems and in Change-Compare problems, is of great significance. Several researchers have showed the importance of other expressions in the statement of problems (De Corte and Verschaffel, 1991; Teubal and Nesher, 1991).
In our research, Combine problems (where the addition of two partial states equals the total state) are not considered, because our interest is really focussed on the contrasts between different expressions in a same numerical situation. Of course, other classes of problems have certainly been described in the literature on this subject (Bruno and Martinon, 1996, 1997).
There are many contexts within which it is possible to state additive problems, such as: temperature, chronology, length, etc. In our research, however, we have preferred to analyze all of them within the same "having money" context ("Juan has 2 pesetas", "Juan has earned 2 pesetas", etc.), so as to fix the context variable as a standard and also because we consider students are generally more familiar with this approach. …