The importance of the statement in addition and subtraction word problems.

1. Introduction

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?

2.1 Compare

We consider two states s ("J has 2 pta") and b ("P has 5 pta"). Both of these interrelate through difference d = b - s ("P has 3 pta more than J"). We can therefore say this type of story does have the scheme s + d = b, where, to avoid confusion of classification, we will take it that d > 0; that is, s is the small state and b is the big state (s < b). We say that an additive history (and its associated problems) is a Compare history if its scheme is s + d = b and the difference d is expressed in some of the following ways:

* More: when it is said how much the big state is "more than" the small state

* Less: when it is said how much the small state is "less than" the big state

Example: "J has 2 pta and P has 5 pta"; the comparison may be expressed in the following ways: "P has 3 pta more than J" (More) and "J has 3 pta less than P" (Less). There are three problems according to the unknown: s (small state), d (difference) or b (big state). For example: "J has 3 pta less than P. P has 5 pta. How many pta does J have?"

2.2 Equalize

In the stories with scheme s + d = b the difference d may also be expressed in the following ways:

* Add on: when it is said how much the small state must "increase" to equalize the big state.

* Take away: when it is said how much the big state must "decrease" to equalize the small state.

Example: "J has 2 pta and P has 5 pta". Then, "If J earns 3 pta, then he will have the same as P" (Add on) and "If P loses 3 pta, then he will have the same as J" (Take away). The stories of this type are referred to as Equalize. Also, in this case, there are three problems associated with each story. For example: "If P earns 2 pta, then he will have the same as R. R has 6 pta. How many pta does P have?"

2.3 Change

Now we consider a start state s ("this morning J had 2 pta"), time goes by and we have the end state e ("J has now 5 pta"), and there is a variation v = e - s ("J has earned 3 pta"). We may say this type of story has scheme s + v = e. If the variation v is expressed in simple form ("J has earned", "J has lost"), then we say that this is a Change story. Consider the following example: "In the morning, J had 2 pta and in the evening he has 5 pta". Then we tell: "J earned 3 pta throughout the day" (Simple). According to the sign of v, we will refer to this as a Change (Increase) when v > 0 ("J has earned 3 pta") or as a Change (Decrease) when v < 0 ("J has lost 3 pta"). In this class of problems, the unknown may be s (start state), v (variation) or e (end state). Example: "J had 5 pta before and he has now 2 pta. How many pta did he lose?"

2.4 Change Compare

A story is said to be of class Change-Compare if it is an additive situation with scheme s + v = e and the variation v is expressed using the words more than or less than, which are used in the expression of the comparison in the stories of the Compare type. Example: "In the evening, J has 3 pta more than he had in the morning" (More) and "In the morning, J had 3 pta less than he has in the evening" (Less). Of course, there are three problems associated with each story, depending on the unknown (start state, variation, end state). Now we also distinguish between classes Change-Compare (Increase) and Change-Compare (Decrease).

2.5 Order of presentation of numbers in the statement

In the problems with scheme x + y = z, the data in the statement can be presented following the x, y, z order (x,y; x,z; y,z) or the opposite order (y,x; z,x; z,y). We have identified the problems with an 'opposite order' with an asterisk *. For instance, the following problems have a different data order:

* P had 6 pta before and he has now 4 pta. How many pta more than he has now did he have before?,

* P has now 4 pta and he had 6 pta before. How many pta more than he has now did he have before?

2.6 Types of problems

Each problem will be classified according to the following categories:

* Type of story: Compare, Equalize, Change, and Change-Compare

* Type of expression: more, less, add on, take away.

* Unknown: start state, variation, end state, small state, difference, big state.

* Order of the data: scheme order, opposite order (*).

The 39 problems considered in our research study are listed in the Appendix and the terminology used is contained in Table 2. Of the 39 problems that we have considered, 9 are of the class Change, 6 of Equalize, 6 of Compare and 18 of Change-Compare.

In the following sections we analyze some characteristics of the problems that we consider relevant in the solution of problems by students.

2.7 Characteristic 1: Inconsistent language

In the literature on this subject, it is usually stated that a problem has inconsistent language (1) when the "key words" used in the statement might be considered to suggest a different calculation to that which really applies. For instance, in the problem:

* Compare 2: T has 2 pta. T has 3 pta less than E. How many pta does E have? We must add 2 + 3, but the expression "less than" may lead certain students to subtract. Similarly, in the problem:

* Change 5: J has earned 3 pta and he has now 5 pta. How many pta did he have before?

We must subtract 5 - 3, even though the term "earned" may suggest addition. That is to say: a problem is worded with inconsistent language when addition is required and expressions such as "less than" or "to lose" are used, or if subtraction is necessary and the expressions "more than" or "to earn" are used. (Problems in which inconsistent language is used are shown in table 2).

2.8 Characteristic R: the referent is the unknown

In the Compare problems and the Change-Compare problems the relationship between the two states is expressed by taking one of them as the referent. For instance, in the Compare problem:

* Compare 1: T has 2 pta. M has 5 pta more than T. How many pta does M have? The referent is what T has. If the comparison is expressed with "less", as in:

* Compare 2: T has 2 pta. T has 3 pta less than M. How many pta does M have? The referent is now what M has, which coincides with the unknown. In the cases in which the referent coincides with the unknown we note by R. In Change-Compare problems there is also a referent. In the problem:

* Change-Compare 1.2: Before, A had 3 pta. Before he had 5 pta less than he has now. How many pta does he have now?

the referent is the number of pta he has now. In the Change problems, we have not considered a referent to exist. The R-problems of our research are shown in Table 2.

2.9 Characteristic IR: Inconsistent language and the unknown is the referent

We apply the symbol "IR" when the I and R characteristics are both present. It should be noted that this only appears in certain Change-Compare and Compare problems (see Table 3). In these problems, and only in them, the referee is the known state, whereas the unknown state is the referent. In the example Change-Compare 5.5, the referee is what L has now and the referent is what L had before:

* Change-Compare 5.5: Now, L has 3 pta more than he had before. Now, he has 5 pta. How many pta did he have before?

Let us consider the above and the following problems:

* Change-Compare 2.1: Before, P had 8 pta. Before he had 3 pta more than he has now. How many pta does he have now?

* Compare 2: T has 2 pta. T has 5 pta less than E. How many pta does E have?

* Compare 5: T has 5 pta more than M. T has 7 pta. How many pta does M have?

In the Change-Compare 2.1 and Compare 2 problems, the above condition is expressed in a strong from (s) because of the "repetitive" expressions "Before ... Before ... now ... now" and "T ... T ... E ... E", whereas in the Change-Compare 5.5 and Compare 5 problems it appears in a weak form (w) "Now ... before. Now ... before" and "T ... M ... T ... M". We believe the distinction between the strong and weak forms to be of importance because we also consider that strong form statements are more difficult for students to understand than weak form statements.

It should be noted that if the order of the numbers presented in the statement of Change-Compare 2.1 [IR(s)] is changed, the problem becomes Change-Compare 2.1* [IR(w)]:

* Change-Compare 2.1*. Before R had 3 pta more than he has now. Before, he had 8 pta. How many pta does he have now?

If we change the order of presentation of numbers in the statement, then a problem with characteristic IR(s) is converted into a problem with characteristic IR(w) and vice versa.

We should mention that some authors (e.g., Verschaffel, 1994) say that a problem is of "inconsistent language" when it has the IR-characteristic, which is yet a stricter use of that terminology.

3. The research study

A research study was carried out among several groups of students to contrast Change-Compare problems with Change, Equalize and Compare problems, and to establish the influence of the expression of the variation and the difference, as well as the effect of the problem characteristics. The results of the research study are described in this paper.

Several groups of students of Third, Fourth, Fifth and Sixth Year of Primary Education in Spain (aged 8-12) were asked to complete different written tests that contained the 39 additive problems referred to in the previous section. The groups of students were selected starting from the third level because the students had previously worked with these problems (on the first and second level).

A total of 267 students of two State Schools (S1 and S2) located in the suburbs of Santa Cruz de Tenerife (Canary Islands, Spain), took part in our experiment, and formed 15 different groups (G1, G2, ... G15).

The large number of problems considered in this experience suggested to distribute them into different tests with an acceptable number of problems on each one, in such a way that the students were able to answer them during one session. The research study covered 10 different tests (T1, T2, T3, ... T10) of six problems each, details of which are given in Table 3. The tests were randomly distributed in each group and the students were asked only to answer one test each.

Table 4 shows the contents of the tests. Please note that the T1 to T6 tests do not contain *-problems and that this type was only included in the T7 to T10 tests. It was intended that a test should contain problems to allow diverse aspects to be examined simultaneously.

For instance, Test 1 included the following problems

T1: Change 5, Change-Compare 2.1, Change-Compare 5.5, Change-Compare 5.6, Equalize 5, Compare 5.

Problems Change 5, Change-Compare 5.5 and Change-Compare 5.6 only differ in the expression of the Variation; analogously, with Equalize 5 and Compare 5 problems. On the other hand, the Equalize 5 and Change-Compare 5.5 problems only differ in their scheme s + d = b or s + v = e. We also wished to examine the contrast between the Change 2 and the Change 5 problems. In paragraph 4.7, advantage was taken of the fact that certain sets of problems appeared in the same test.

In order to avoid possible influences through the order in which problems are set, several different formats were used for each individual test, changing the order of problems.

The students' usual answer was either to write an operation (addition, subtraction, multiplication or division), or simply to write the solution to the problem without giving details of their operations. Their answers were classified as right, wrong or blank.

4. Results and discussion

Table 5 shows the percentage of general success achieved for each of the problems, the number of students to whom it was put and the characteristics of such problems, with students of all levels and groups being evaluated together.

It may be readily appreciated that the results are of a very diverse nature and a detailed study, based on problem characteristics, is required.

4.1 General results

It is evident that we cannot establish a strict level of difficulty between the four classes of problems under consideration. The relationship of the I-, R- and IR-characteristics, especially the latter, with the greatest problem-solving difficulty, is evident in Table 5. It can be appreciated that IR(s)-problems are generally those with a lower success percentage, particularly the Change-Compare problems; particularly, Change-Compare 1.2 (37%) and Change-Compare 2.1 (29%) problems. In the IR(w)-problems, the success rates are higher: Change-Compare 5.5 (65%) and Change-Compare 6.6 (47%). The greater influence of the strong form on the results may be appreciated more clearly by reference to the (*)-problems: Change-Compare 2.1 [IR(s)] has a 29% success percentage and Change-Compare 2.1 [IR(w)] has a 45%; similarly, problem Change-Compare 6.6* [IR(s)], with 21% as against 47% in the Change-Compare 6.6 [IR(w)] problem. The influence of the IR-characteristic on the Compare problems is not so marked as that in the Change-Compare problems.

4.2 Data order

With regard to the (*)-problems, we can establish whether the order in which data is presented has any influence on the success rates. A quick reference to Table 5 clearly shows that this is important with the Change-Compare problems but of little or no consequence where Change problems are concerned. We have already pointed out that the order in which the data is expressed in IR problems causes the statement to adopt a strong or weak form of expression, with a direct repercussion on the success rates. Such results lead us to believe that the Change-Compare 1.2* and Compare 2* problems (which, we repeat, have not been studied in the present work), both IR(w), are more easily solved by students than their IR(s) counterparts: the Change-Compare 1.2 and Compare 2 problems, respectively. Similarly, we think that the Change-Compare 5.5* and Compare 5* problems, both IR(s), are found to be more difficult than their IR(w) counterparts: Change-Compare 5.5 and Compare 5, respectively.

4.3 Change

Problems under this heading are clearly those with the best problem solving success rates, all between 75% and 100% (see Table 5). It may be that, because more attention is given at school to this type of problem and to those of the "states' combination" type, such success rates are so positively influenced. The lower values in the 75%-100% interval are related to 1-characteristic problems, actually between 75% and 81%, while the rest are between 91% and 100%. Please note that problems Change 6* (81%), Change 4* (94%) and Change 2 (94%) figured exclusively in Test 10, which corroborates our opinion that problems worded with inconsistent language "I", are those where students encounter the greatest difficulty. In the comparison of our results with those of Martinez and Aguilar (1996), their investigation shows that the worst results were also related to problems containing the 1-characteristic. It should be noted, however, that problems with the IR or simply R characteristics do not exist in the class of the Change problems.

4.4 Change Compare

As may be seen in Table 5, three of the problems in this category show success rates that are clearly inferior to all the others in this research study: Change Compare 1.2 (37%), Change-Compare 2.1 (29%) and Change Compare 6.6* (21%), which are the three IR(s)-problems. Low success rates also affect three other problems with the IR(w)-characteristic: Change-Compare 5.5 (65%), Change-Compare 6.6 (47%) and Change-Compare 2.1* (45%).

As to the problem-solving difficulty levels of the Change-Compare (Increase) problems, we have: the IR(s) with a 37% success percentage, the IR(w) with 65%, followed by the one with I-characteristic, Change-Compare 3.3, with 78%, and lastly the problems without a specific characteristic, between 80% and 93%. The Change-Compare (Decrease) problems, however, show a different pattern, the IR(s) type proving to be the most difficult, with success results of 21% and 29%, followed by the IR(w), with results of 45% and 47%. Nevertheless, there are two I-problems, Change-Compare 4.3 (95%) and Change-Compare 4.3* (64%), which, of all problems without the IR characteristic, have the lowest and highest success percentage figures, respectively, something we cannot clearly explain.

4.5 Compare

The two problems with the lowest success rates do possess the IR-characteristic: Compare 5 [IR(w)] 63% and Compare 2 [IR(s)] 59%. They also represent by far the lowest two success rates in the study of Martinez and Aguilar (1996). Of the two problems of unknown Diff, the lowest percentage was in respect of Compare 3 (74%), an I-problem (See Table 5).

4.6 Equalize

As shown in Table 5, the two problems, Equalize 6 and Equalize 1, with the unknown as referent, do not have low success rates. The lowest percentages are shown against Equalize 2 (50%), Equalize 5 (57%) and Equalize 3 (75%), which are the three problems with the I-characteristic, the same result having been obtained by Martinez and Aguilar (1996). In this class of problems, therefore, the I-characteristic makes a problem more difficult to solve than the R-characteristic.

4.7 Similar problems

For the problems Change-Compare 5.6 (80%), Change-Compare 2.2* (79%) and Compare 6 (80%), not only were similar success rates achieved but their respective statements may be obtained from each other by interchanging their words. Thus, if in the case of Change-Compare 5.6:

* Change-Compare 5.6. Before E had 4 pta less than he has now. Now, he has 9 pta. How many pta did he have before?

we replace "before" and "now" by "now" and "before", respectively, and verbal forms are appropriately modified, Change-Compare 5.6 becomes Change-Compare 2.2*

* Change-Compare 2.2*. Now J has 2 pta less than he had before. Before, he had 6 pta. How many pta does he have now?

If with Change-Compare 2.2*, "now" and "before" are replaced by "R" and "L", respectively, it becomes Compare 6:

* Compare 6. R has 3 pta less than L. L has 5 pta. How many pta does R have?

Similar possibilities exist with other sets of problems, as shown in Table 6, although the success percentage values are not so close to each other as in the previous example. Nevertheless, each of the different sets-of-three problems was included with the same individual test and the results confirm the coincidence.

Considering (*)-problems of Change (Increase) and of Compare (not the subject of the present research), we can take it for granted that the sets-of-three problems shown in Table 7 below would have similar success percentage values.

Also, the similar results obtained in the Change-Compare 6.6 (47%) and Change-Compare 2.1* (45%) problems, as also in the Change-Compare 6.6* (21%) and Change-Compare 2.1 (29%) problems, are certainly remarkable. In the following statements:

* Change-Compare 6.6. Now, M has 4 pta less than he had before. Now, he has 5 pta. How many pta did he have before?

* Change-Compare 2.1*. Before R had 3 pta more than he has now. Before, he had 8 pta. How many pta does he have now?

if we replace the words "now" by "before" and "less" by "more", one type of problem is converted to the other. The same occurs in the following example:

* Change-Compare 6.6*. Now, P has 5 pta. Now, he has 4 pta less than he had before. How many pta did he have before?

* Change-Compare 2.1. Before, P had 8 pta. Before he had 3 pta more than he has now. How many pta does he have now?

Similar comments may also be made about other sets of problems. It seems, therefore, that all problems with similar statements will normally be solved with near-equal success results.

5. Conclusions

The main object of the research presented in this paper is to highlight that the manner in which certain problem statements are expressed certainly influences problem-solving success results.

A simple additive situation can be expressed in different ways. If the situation is associated to a scheme s + v = e (start state + variation = end state), there are three ways of expression of the variation (simple, more than, less than); the results of our research show that the kind of expression used can affect the level of success. Analogously, if the scheme is s + d = b (small state + difference = big state), then there are four ways of expression of the difference (more than, less than, take away, add on), and the expression used in the statement is related with the difficulty of the problem.

In order to confirm our idea concerning the importance of the expression, we have taken four classes of additive problems with positive numbers, three of which (Change, Compare and Equalize) were previously quite well-known, and the fourth class (Change-Compare) has been investigated in our research for the very first time. The Change and Change-Compare problems have the same scheme s + v = e, but the expression of the variation is different; also the Compare problems and the Equalize problems have the scheme s + d = b, but the difference is expressed in different ways. Moreover, the expressions for the variation and the difference are similar in the Change and Equalize problems, and also in Change-Compare and Compare problems.

With the object of investigating what influence the manner in which a problem is expressed may have, certain problem characteristics were considered. Characteristic I is used to indicate "inconsistent language" in the expression of a problem that might suggest addition when in fact the correct action to be taken is to subtract, or to subtract when addition should apply. Characteristic R will also apply when the referent is the unknown. The results of our investigation prove that the problem-solving results are adversely affected by both the characteristics, although when both coincide (IR) the difficulty created is much greater. This is particularly true when the order in which the data is expressed "forces" the characteristic IR to adopt the strong form: IR(s).

After the detailed analysis carried out, we can establish that Change-Compare problems have a particular condition of their own. On the one hand, Change problems do not have the IR-characteristic, whereas it certainly appears in Change-Compare problems, therefore, making them more difficult to solve. On the other hand, Change-Compare problems differ greatly from Compare ones, because Change-Compare problems with the IR-characteristic are among those with the lowest problem-solving success rates. There are others, however, with the I-characteristic, having similar success rates for both classes. So, it can be said that the IR-characteristic is a particularly negative condition with Change-Compare problems but of less consequence in Compare or Equalize problems.

The fact that some of the problems show a low percent of success does not imply that they should be excluded from the teaching. On the contrary, they should have a careful didactical treatment at the school through methodologies giving advantage to the understanding of the statement. In the context of the problems to which we have referred in the present investigation, we are sure an understanding of the numerical situation expressed in the statement of the problem, providing the student with some kind of mental picture, is of great importance for achievement of the correct solution. Therefore, we believe the poorer success rates that correspond to certain problems simply result from statements with numerical information that are difficult to grasp; so the student in his/her reply, if any, just "processes" the data in what he believes to be a rational way (using key words). This fact had already been established with regard to the Compare problems but our research has nevertheless allowed us to extend the results to the Change-Compare problems.

It is really necessary to complement our investigation with certain cognitive aspects of student in the problem-solution process. Specifically, it will be interesting to ascertain what conceptual scheme is used by students and whether they in fact utilize the same scheme for the solution of Change-Compare as for the Change problems, or, contrarily, whether they just use the same as for the Compare problems. Clarification of this particular point would be vital in establishing exactly whether the dominant influence in the solution of a problem is the structure (the scheme) of the problem, or the way in which the problem statement is actually expressed.

It is possible that a "compatibility hypothesis", similar to that of Lewis and Mayer, in the solution of Compare problems (Verschaffel, 1994), may be considered applicable to some students, who would evidently prefer the statement of a Change-Compare problem to be expressed without the IR-characteristic. When it is present, however, they could "mentally reorganize" the problem statement to exclude the characteristic in question and exchange a More with a Less one. A further research is required to confirm this hypothesis, and we are already planning to carry that out.

Our investigation has produced certain results that should favor education. From what we have stated, it is evident that it is essential that the student clearly understands the statement of a problem so that he/she is able to create a mental picture of the numerical situations involved. In this respect, it has been showed in several investigations that the use of diagrams and graphics, or mathematics manipulative teaching material, can be of invaluable assistance in the comprehension of problems (De Corte, 1993; Riley et al., 1983). It is also beneficial for students to get directly involved with the statements of problems, either by changing statements previously given to them in their own words (Verschaffel, 1994), or just making up completely fresh problems themselves (Rudnitsky et al., 1995).

Several authors (such as Rudnitsky et al., 1995) consider it convenient that the knowledge of the different classes of problems form part of the normal school education curriculum, and that students should be able to identify them as Change, Equalize or Compare. If we accept this fact, it is thought that working with students about the four classes of problems we have discussed here will improve their understanding of problematic situations and, therefore, improve their problem-solving ability. We are also sure the distinction we have made between the scheme of the problem (s + v = e; s + d = b) and the ways in which the relationship between the two states (Variation and Difference) may be expressed, will both contribute to achieve this purpose.

Finally, the conclusions we have drawn from our work actually strengthen those already described in various publications on the importance of the particular attention that textbook authors should pay to the expressions employed in the enunciation of problems. School teachers should also be conscious of the relevance of the expression in problem enunciation, not only when the problem is being solved but also during subsequent analysis of the students' results.

Appendix: Problem Statements

Note: In this paper, we have forced the usual syntactic order in English in order to show the syntactic pattern used in the statements of the problems in Spanish.

Change

* Change 1: Before, A had 4 pta. Then he earned 3 pta. How many pta does he have now?

* Change 2: Before, J had 9 pta. Then he lost 4 pta. How many pta does he have now?

* Change 2*: Before losing 4 pta, J had 9 pta. How many pta does he have now?

* Change 3: J had 2 pta before and he has now 5 pta. How many pta did he earn?

* Change 4: J had 5 pta before and he has now 2 pta. How many pta did he lose?

* Change 4*: E has now 2 pta and he had 5 pta before. How many pta did he lose?

* Change 5: J has earned 3 pta and he has now 5 pta. How many pta did he have before?

* Change 6: After losing 3 pta, R has now 4 pta. How many pta did he have before?

* Change 6*: R has now 4 pta, after losing 3 pta. How many pta did he have before?

Change-Compare

* Change-Compare 1.1: Before, P had 5 pta. Now, he has 4 pta more than he had before. How many pta does he have now?

* Change-Compare 1.2: Before, A had 3 pta. Before, he had 5 pta less than he has now. How many pta does he have now?

* Change-Compare 2.1: Before, P had 8 pta. Before, he had 3 pta more than he has now. How many pta does he have now?

* Change-Compare 2.1*: Before, R had 3 pta more than he has now. Before, he had 8 pta. How many pta does he have now?

* Change-Compare 2.2: Before, A had 6 pta. Now, he has 2 pta less than he had before. How many pta does he have now?

* Change-Compare 2.2*: Now, J has 2 pta less than he had before. Before, he had 6 pta. How many pta does he have now?

* Change-Compare 3.3: L had 4 pta before and now he has 7 pta. How many pta more than he had before does he have now?

* Change-Compare 3.4: L had 5 pta before and he has now 9 pta. How many pta less than he has now did he have before?

* Change-Compare 4.3: P had 6 pta before and he has now 4 pta. How many pta more than he has now did he have before?

* Change-Compare 4.3*: P has now 4 pta and he had 6 pta before. How many pta more than he has now did he have before?

* Change-Compare 4.4: M had 9 pta before and he has now 5 pta. How many pta less than he had before does M have now?

* Change-Compare 4.4*: M has now 5 pta and he had 9 pta before. How many pta less than he had before does M have now?

* Change-Compare 5.5: Now, L has 3 pta more than he had before. Now, he has 5 pta. How many pta did he have before?

* Change-Compare 5.6: Before, E had 4 pta less than he has now. Now, he has 9 pta. How many pta did he have before?

* Change-Compare 6.5: Before, L had 3 pta more than he has now. Now, he has 2 pta. How many pta did he have before?

* Change-Compare 6.5*: Now, J has 2 pta. Before, he had 3 pta more than he has now. How many pta did he have before?

* Change-Compare 6.6: Now, M has 4 pta less than he had before. Now, he has 5 pta. How many pta did he have before?

* Change-Compare 6.6*: Now, P has 5 pta. Now, he has 4 pta less than he had before. How many pta did he have before?

Equalize

* Equalize 1: R has 4 pta. If R earns 2 pta, then he will have the same as P. How many pta does P have?

* Equalize 2: A has 4 pta. If C loses 2 pta, then he will have the same as A. How many pta does C have?

* Equalize 3: T has 3 pta and M has 8 pta. How many pta does T have to earn to have the same as M?

* Equalize 4: C has 2 pta and A has 7 pta. How many pta does A have to lose to have the same as C?

* Equalize 5: If P earns 2 pta, then he will have the same as R. R has 6 pta. How many pta does P have?

* Equalize 6: If C loses 5 pta, then he will have the same as A. C has 7 pta. How many pta does A have?

Compare

* Compare 1: T has 2 pta. M has 5 pta more than T. How many pta does M have?

* Compare 2: T has 2 pta. T has 5 pta less than E. How many pta does E have?

* Compare 3: R has 2 pta and A has 7 pta. How many pta more than R does A have?

* Compare 4: P has 4 pta and A has 6 pta. How many pta less than A does P have?

* Compare 5: T has 5 pta more than M. T has 7 pta. How many pta does M have?

* Compare 6: R has 3 pta less than L. L has 5 pta. How many pta does R have?

Table 1. The four classes of problems analyzed in our research study
(pta = pesetas)

             Scheme: s + v = e                  Scheme: s + d = b

Expression:  Change                             Equalize
Simple       Before Juan had 4 pta.             Juan has 4 pta.
             Then he earned 3 pta.              If Juan earns 2 pta,
             How many pta does he have now?     then he will have the
                                                same as Pedro.
                                                How many pta does Pedro
                                                have?
Expression:  Change-Compare                     Compare
More/Less    Before Juan had 4 pta.             Juan has 4 pta.
             Now he has 3 pta more than he had  Pedro has 5 pta more
             before.                            than Juan.
             How many pta does he have now?     How many pta does Pedro
                                                have?

Table 2. Types of problems considered in this research study

Type of problems     Expression  Unknown     Sign of    Characteristic
                                             variation

Change
Change 1             Add on      End         Increase
Change 2             Take away   End         Decrease
Change 2*
Change 3             Add on      Variation   Increase   I
Change 4             Take away   Variation   Decrease
Change 4*
Change 5             Add on      Start       Increase   I
Change 6             Take away   Start       Decrease   I
Change 6*                                               I
Compare
Compare 1            More        Big
Compare 2            Less        Big                    IR(s)
Compare 3            More        Difference             I
Compare 4            Less        Difference
Compare 5            More        Small                  IR(w)
Compare 6            Less        Small
Equalize
Equalize 1           Add on      Big                    R
Equalize 2           Take away   Big                    I
Equalize 3           Add on      Difference             I
Equalize 4           Take away   Difference
Equalize 5           Add on      Small                  I
Equalize 6           Take away   Small                  R
Change-Compare
Change-Compare 1.1   More        End         Increase
Change-Compare 1.2   Less        End         Increase   IR(s)
Change-Compare 2.1   More        End         Decrease   IR(s)
Change-Compare 2.1*                                     IR(w)
Change-Compare 2.2   Less        End         Decrease
Change-Compare 2.2*
Change-Compare 3.3   More        Variation   Increase   I
Change-Compare 3.4   Less        Variation   Increase
Change-Compare 4.3   More        Variation   Decrease   I
Change-Compare 4.3*                                     I
Change-Compare 4.4   Less        Variation   Decrease
Change-Compare 4.4*
Change-Compare 5.5   More        Start       Increase   IR(w)
Change-Compare 5.6   Less        Start       Increase
Change-Compare 6.5   More        Start       Decrease
Change-Compare 6.5*
Change-Compare 6.6   Less        Start       Decrease   IR(w)
Change-Compare 6.6*                                     IR(s)

Table 3. Distribution of students by levels, groups, schools and tests
proposed

Level                  Age    S   G    N    T1  T2  T3  T4  T5  T6  T7

3[degrees]-6[degrees]   8-12           267  30  32  32  32  32  30  22
3[degrees]              8-9             70   4   8   8   8   8   8   8
                              1   G1    23   3   4   4   4   4   4
                              1   G2    21   1   4   4   4   4   4
                              2   G3    15                           5
                              2   G4    11                           3
4[degrees]              9-10            59   8   8   8   9   8  10   2
                              1   G5    17   4   4   3   4   1   1
                              1   G6    24   4   4   4   4   3   5
                              1   G7    10   -   -   1   1   4   4
                              2   G8     8                           2
5[degrees]             10-11            66   8   8   8   7   8   6   6
                              2   G9    23   4   4   4   4   4   3
                              2   G10   22   4   4   4   3   4   3
                              2   G11   21                           6
6[degrees]             11-12            72  10   8   8   8   8   6   6
                              2   G12   18   4   4   4   4   2   -
                              2   G13   15   4   4   3   -   2   2
                              2   G14   15   2   -   1   4   4   4
                              2   G15   24                           6

Level                  Age    S   G    N    T8  T9  T10

3[degrees]-6[degrees]   8-12           267  19  22   16
3[degrees]              8-9             70   6   6    6
                              1   G1    23
                              1   G2    21
                              2   G3    15   2   4    4
                              2   G4    11   4   2    2
4[degrees]              9-10            59   1   5    -
                              1   G5    17
                              1   G6    24
                              1   G7    10
                              2   G8     8   1   5    -
5[degrees]             10-11            66   6   5    4
                              2   G9    23
                              2   G10   22
                              2   G11   21   6   5    4
6[degrees]             11-12            72   6   6    6
                              2   G12   18
                              2   G13   15
                              2   G14   15
                              2   G15   24   6   6    6

S = School; G = Group; N = Number of students

Table 4. Contents of the tests

Test                       Types of problems

 1    Change 5             Change-Compare 2.1  Change-Compare 5.5
      Change-Compare 5.6   Equalize 5          Compare 5
 2    Change 1             Change-Compare 6.5  Change-Compare 1.1
      Change-Compare 1.2   Equalize 1          Compare 1
 3    Change 2             Change-Compare 5.6  Change-Compare 2.2
      Change-Compare 2.1   Equalize 6          Compare 6
 4    Change 6             Change-Compare 1.2  Change-Compare 6.6
      Change-Compare 6.5   Equalize 2          Compare 2
 5    Change 3             Change-Compare 4.3  Change-Compare 3.3
      Change-Compare 3.4   Equalize 3          Compare 3
 6    Change 4             Change-Compare 3.4  Change-Compare 3.4
      Change-Compare 3.3   Equalize 4          Compare 4
 7    Change-Compare 6.5*  Change-Compare 5.5  Change-Compare 2.1*
      Change-Compare 1.1   Compare 5           Compare 1
 8    Change-Compare 2.2*  Change-Compare 5.6  Change-Compare 6.6*
      Change-Compare 1.2   Compare 6           Compare 2
 9    Change-Compare 4.4*  Change-Compare 3.3  Change-Compare 4.3*
      Change-Compare 3.4   Compare 3           Compare 4
10    Change 6*            Change 4*           Change 2*
      Equalize 6           Equalize 4          Equalize 2

Table 5. Success rates, number of students and characteristics of the
problems

            %         N   Char   %         N   Char  %         N   Char

Change      Change 1             Change 3            Change 5
            94        32          75       32  I     77        30  I
            Change 2             Change 4            Change 6
            91        32         100       30        78        32  I
* problem   94        16          94       16        81        16  I
Change-     Change-Compare 1.1   Change-Compare 3.3  Change-Compare 5.5
Compare
            93        54          78       54  I     65        52  IR(w)
            Change-Compare 1.2   Change-Compare 3.4  Change-Compare 5.6
            37        83  IR(s)   90       84        80        81
            Change-Compare 2.1   Change-Compare 4.3  Change-Compare 6.5
            29        62  IR(s)   95       62  I     70        64
* problem   45        22  IR(w)   64       22  I     91        22
            Change-Compare 2.2   Change-Compare 4.4  Change-Compare 6.6
            84        32          94       30        47        32  IR(w)
* problem   79        19          82       22        21        19  IR(s)
Equalize    Equalize 1           Equalize 3          Equalize 5
            91        32  R       75       32  I     57        30  I
            Equalize 2           Equalize 4          Equalize 6
            50        48  I       98       46        85        48  R
Compare     Compare 1            Compare 3           Compare 5
            80        54          74       54  I     63        52  IR(w)
            Compare 2            Compare 4           Compare 6
            59        51  IR(s)   89       52        80        51

% = Success; N = Number of students; Char = Characteristic

Table 6. Sets-of-three similar problems

                A   T                        A   T              A   T

Change-Compare
  5.5           65  55  Change-Compare 2.1*  45  45  Compare 5  63  68
Change-Compare
  3.3           78  73  Change-Compare 4.3*  64  64  Compare 3  74  68
Change-Compare
  1.1           93  95  Change-Compare 6.5*  91  91  Compare 1  80  95
Change-Compare
  5.6           80  79  Change-Compare 2.2*  79  79  Compare 6  80  74
Change-Compare
  3.4           90  91  Change-Compare 4.4*  82  82  Compare 4  89  86
Change-Compare
  1.2           37  32  Change-Compare 6.6*  21  21  Compare 2  59  42

A = Average for all tests;
T = Average for individual tests with the sets-of-three problems in the
same row

Table 7. Possible sets-of-three similar problems

Change-Compare 5.5*  Change-Compare 2.1  Compare 5*
Change-Compare 3.3*  Change-Compare 4.3  Compare 3*
Change-Compare 1.1*  Change-Compare 6.5  Compare 1*
Change-Compare 5.6*  Change-Compare 2.2  Compare 6*
Change-Compare 3.4*  Change-Compare 4.4  Compare 4*
Change-Compare 1.2*  Change-Compare 6.6  Compare 2*

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Alicia Bruno and Antonio Martinon

University of La Laguna, Spain

Fidela Velazquez

Secondary Institut San Hermenegildo, Spain