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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. Please note that in order to simplify, we will now use "pta" instead of "pesetas" and the initials J, P, E, T for persons' names.
In each simple additive situation there are three problems, depending on the unknown (these are described in the next section). The unknown also has an important influence on the problem-solving result (Carpenter and Moser, 1982; Riley et al., 1983).
In our research involving students in Third, Fourth, Fifth and Sixth Year of Primary Education in Spain (aged 8-12), we have analyzed the levels of problem-solving difficulty of the four classes of problems and have evaluated the results in accordance with certain characteristics which we consider certainly influence the problems. The characteristics related to the expression of the difference and the variation, which will be described in more detail in the following section, are identified as follows: "I" = use of "inconsistent language" and "R" = the referent is the unknown. The combination of these two characteristics (IR) may appear in a strong form (s) or in a weak form (w), according to the order of the data in the statement. The problems identified with the characteristic IR(s) are in fact those with a lower problem-solving success rate and appear in Change Compare and Compare problems. Change-Compare problems, therefore, have certain factors that distinguish them from Change problems and, certainly, from Compare and Equalize problems.
Student problem-solving success rates, for the problems that were set, certainly confirm our belief that they are influenced by the way in which the variation and the difference are expressed and indeed the order of the data itself. In our opinion, the reason is that there are certain ways of expressing variation and difference, or the sequence of the data as such, that create greater difficulty for the student when trying to understand the statement of the problem, or to visualize the numerical situation involved, really making a solution very difficult. Our explanation is therefore based (in the case of these simple additive problems with positive numbers) on the fact that levels of problem-solving success are directly related to the degree to which the statement of the problem is clearly understood.
It should be noted that in this paper we have "forced" the usual syntactic order in English in order to maintain the syntactic pattern used in the statements of the problems is Spanish: "Ahora Pedro tiene 5 pesetas. Ahora tiene 4 pesetas menos que antes. ?Cuantas pesetas tenia antes?" ("Pedro has now 5 pesetas. He has now 4 pesetas less than he had before. How many pesetas did he have before?)
2. Types of problems
In this section, we introduce the terminology used, describe the types of additive problems that were set for the students and highlight the characteristics of the problems we consider more relevant.
Before we refer to such additive problems, we will discuss the additive story (Rudnisky et al., 1995) or additive situation, in which a situation involving the addition or subtraction of two numbers is described. In our research, simple additive stories with positive numbers are considered, which are those with the addition x + y = z or the subtraction x - y = z, where x, y, z are all positive. We consider four classes of additive stories and, consequently, four classes of additive problems: Compare, Equalize, Change and Change-Compare, as described in the following paragraphs.
We can have three types of problems associated with a story of this kind, depending on the unknown. The story "J has 2 pta and P has 5 pta, so P has 3 pta more than J," gives rise to the following three problems:
* J has 2 pta and P has 5 pta. How many pta more than J does P have?
* J has 2 pta P has 3 pta more than J. How many pta does P have?
* P has 3 pta more than J. P has.5 pta. How many pta does J have?
We consider two states s ("J has 2 pta") …