1994

THE ROLE OF SCHEMES IN TWO STEP PROBLEMS 

Analysis and Research findings

Pearla Nesher - The university of Haifa

Sara Hershkovitz - Centre for Educational Technology

The work reported herein is an attempt to understand the representations involved in solving complex word problems. In particular we will try to substantiate the effect of schemes hypothesized to underlie two-step arithmetic word problems. Our assumption is that a successful solver is directed and constrained by the scheme of the problem. It is with the aid of the scheme that he is able to interpret a given text or imagine a situation model necessary for his computational solution.

Cognitive scientists in recent years have employed several theoretical notions to describe problem representations, such as "frame" (Minsky, 1975; Hayes, 1979), "script" (Schank & Abelson, 1977) or "scheme" (Bobrow & Norman, 1975; Rumelhart, 1975). Frame and script are viewed as a formal representation of a data organized for a contextual, stereotyped situation, based on past experience. Scheme is viewed as a notion related to frame or script, but in a more abstract form. It is detached from contextual details, and stresses the intersection of various contextual cases. Unlike former theoretical notions (script, frame) which are embedded in a specific theoretical context within which they were developed, scheme is a more ambigous term employed in many theoretical frameworks. Our point of departure in using the term scheme" will be McClelland & Rumelhart (1987) description:

"The basic idea is that schemata are data structures for representing the generic concepts stored in memory. There are schemata for generalized concepts underlying objects, situations, events, sequences of events, actions, and sequences of actions. Roughly, schemata are like models of outside world. To process information with the use of schema is to determine which model best fits the incoming information. Ultimately, consistent configurations of schemata are discovered which, in concert offer the best account for the input. This configuration of schemata together constitutes the interpretation of the input" (p.18).

Most researchers of word problem solving in the past tended to emphasize the direct transition (and translation) from the given text to the numerical computation. In recent years, however, studies assume that solving mathematical word problems is a multi-dimensional process, and other related hypothetical concepts were added, such as the "text base", which is an elaboration of the surface structure verbal formulation into propositions and the "situation Model" which is the representation of the situation described in the text (Kintsch, 1986). In this work we would like to stress the centrality of the scheme in the process of representing word problems.

The advantage of describing word problems in terms of schemes is that given a diffused story the solver is able to relate it to a structure. This structure is a well defined in terms of its arguments, the role of the arguments and the interrelations among them. The role of these arguments is described by the semantic relations embedded in the text. Thus, for one familiar with the possible schemes for a certain domain, the scheme enables him to work in a more structured and constrained domain. While works describing the role of schemes in solving one-step additive or multiplicative word problems are numerous: Decorte, E. Verschaffel, L. & Dewin, L (1982), Kintsch and Greeno (1985), Nesher et al. (1982), Nesher (1988), Nesher & Katriel (1977), Riley et al. (1983); they are scare, however, for the more complex word problems (Shalin & Bee, 1985; Greeno, 1987), Reusser (1990). Past research on one-step additive word problems has already noted several schemes such as the "change" scheme, the "part-part-whole" scheme or the "compare" scheme. These schemes were found to be instrumental in explaining the solution processes of various additive problems. Similarly, schemes were hypothesized for multiplication word problems (Bell et al., 1984; Nesher, 1988; Vergnaud, 1988). It is hoped that the effort undertaken in the present study to substantiate schemes in two-step problems will contribute to the understanding of the cognitive processes involved in solving these problems.

In the present work we are dealing with two-step problems which are combinations of the additive and the multiplicative structures, examining several factors: we tried to tackle the relationship between the situation model as facilitated by various contexts; the impact of the logical-mathematical structure as manifested by the mathematical operations needed to solve the problem; and the schemes that denote the semantical set relations that appear in the problems. Additional variables related to the nature of the reference, whether an extensive or intensive quantity (Shcwartz, 1986), were also included in this study. All the above factors were considered simultaneously in a factorial design.,

This paper is organized as follows: Section 1 clarifies the terms to be used in this work. Section 2 characterizes two-step problems. Sections 3 and 4 categorizes two-step problems by operations and by schemes, respectively. Section 5 presents the empirical search for variables affecting performance on two-step problems, in particular the effect of schemes as compared to the effect of mathematical operations. Section 6 summarizes the findings, and Section 7 concludes and presents our final discussion.

1. Clarifyng the terms

To facilitate communication some of the terms employed in this paper will first be explained particularly what is meant by "component", "underlying structure", and "scheme". Let us start with an example of a one-step problem, a problem in which in order to solve it one has to perform one simple mathematical operation:

There are 13 boys and 16 girls in the group. How many children are there in the group?

In a properly formulated one-step problem the text contains three components (propositions), two of which convey the numerical information - complete components - and one which is the question component, in which the numerical information is missing but the description of the set is given - the incomplete component. The three components in the above problem are:

(1) 13 boys in the group (complete component).

(2) 16 girls in the group (complete component).

(3) How many children are there in the group? (incomplete component)

Each component has two basic parts: the number and the description of the quantified objects. In the first component, the number is 13 and the description is "boys in the group". In the second component the number is 16 and the description is "girls in the group". In the third component we are missing the number (which is the target of the problem) but we know its description - "children in the group". Each problem that evokes a binary operation for its solution is comprised, according to our analysis, of two complete components and one incomplete component.

The above information can be depicted in a graphical scheme consisting of three related components (See Figure 1). Note that regardless of the fact that one of the components is an incomplete component (or an unsaturated argument) one can assign to each component its role in the entire scheme, e.g., the two upper boxes represent the subsets, while the bottom box represents the union of these two subsets.

(Figure 1 about here)

We speak of an additive relation rather than addition or subtraction operations, because the identical situation (29 children in a group composed of 13 boys and 16 girls, in this case) can yield three different problems, depending on the incomplete component, and whether we ask of the boys, the girls or the children. For example, Problem 2, a subtraction problem stemming from the same situation:

There are 29 children in the group. 13 of them are boys.

How many girls are there in the group?

Problem 2 contains the same three components as Problem 1, except for the order and the fact that the designation of the twcomplete components and the incomplete component is different. As more extensively analyzed in a one-step word problem , the incomplete component is crucial for understanding that Problem 1 calls for addition and problem 2 calls for subtraction (Nesher & Katriel, 1977). If after the two complete components in Problem 1, the third incomplete component would be "How many more girls are there in the group than boys?" instead of "How many children are there in the group?"), the problem would become entirely different. (comparison rather than joining groups, and subtraction rather than addition). Thus, we regard a simple word problem as a 3-place relation R(a,b,c), where a,b, and c are the three components (propositions) in the text of the problem. We call this 3-place relation "The underlying structure" of the text (or "the structure") which always consists of three components. Note that we speak about underlying components (propositions) and not of three sentences of the text in the surface structure.

A similar analysis will hold for problems having a multiplicative structure, but we will not deal with this here (See Nehser, 1988).

Finally, the term "scheme" as used in this paper, means a composition of two or more structures. It will be further clarified when we further discuss the two-step problems. The characteristics of the schemes for two-step problems were the target of our study.

2. Two-step problems.

Obviuosly, a two-step problem needs two binary operations for a numerical solution of the problem. For example,

A total of 35 flowers are distributed equally among 7 vases. In each vase there are 3 tulips and the rest are roses. How many roses are there in each vase?

The explicit components of this problem are:

1) A total of 35 flowers is distributed equally among vases (complete component).

2) There are 7 vases (complete component).

3) 3 Tulips in each vase (complete component).

4) How many Roses are there in each vase? (incomplete component).

Based on our analysis of one-step problems, we would expect to find in two-step problems two underlying structures, and therefore six components. But so far we have identified in the text only three numbers and four descriptors. How is the rest of the information that enables solving the problem obtained? We claim that each two-step problem contains an additional latent component which can be deduced from the text though it is not mentioned explicitly. In the above example this latent component is "How many flowers are there in each vase?". The latent component in Problem 3 is the result of one binary operation, division - 35:7=?, in the multiplicative structure underlying the problem, and it is also one of the inputs for the second binary operation, subtraction - ?-3=2, which is part of the additive structure. Thus, the latent component is shared by both structures underlying this two-step problem, and serves as the link between the two structures. It is important to notice that the link between the two structures of problem 3 is not explicitly mentioned in the text and the solver needs to deduce it.

To conclude: as expected, a two-step problem has two underlying structures and six components, four of which are explicitly mentioned in the text, and one is latent and serves twice, in a different role in each structure. These are, for example, for

problem 3:

Multiplicative A) There are 35 flowers altogether

structure B) There are 7 vases.

C) How many flowers are there in each vase? (Latent).

Additive D) There are [?] flowers in each vase. (latent)

structure E) In each vase there are 3 Tulips.

F) How many Roses are there in each vase?

The above analysis, and specifically the uncovering of the latent component will serve us in further analysis of two-step problems and in the categorization into distinct problem types.

3. Categorization of two-step problems by operations

The categorization of problems according to the operations they evoke is straightforward. A pioneer work by Gray & Young (1940) has suggested such a categorization, and used it to compare levels of difficulty of two-step problems. In a work by Jerman and Rees (1972), the mathematical operations were also found to be a significant factor in predicting levels of difficulty of word problems. If we restrict ourselves to two binary operation problems and consider all possible combinations of the four basic mathematical operations (addition, subtraction, multiplication and division) we arrive at 16 distinct types of problems. When non-commutative operations are involved, the order of the operations needs to be considered too, to distinguish for example, between a:(b+c) and (b+c):a ; or a:b+c (Note, that while all involve addition and division, they are different combinations of addition and division). The above three phrases denote a more basic distinction, i.e., which operation will be performed first, and which will be the second, thus, taking into account the order of the operations, we arrive at 24 distinct problems.

(Table 1, about here)

As can be noted, we arrive at 24 categories merely on the basis of the operations and their order. However, a lesson from the one-step problems tells us that categorization by operations does not fully explain the difficulty of certain problems. Research findings show that for the same operation ( + or - ) some problems are easier and some harder depending on their scheme's category - whether they are "change" or "compare" problems.

To be consistent with our previous textual analysis our basic unit of analysis should be a complete structure (a 3-place relation) and not an operation. Thus, while Gray has four basic building blocks (the four operations ( +, -, x, : ) for his elementary two-step problems, we propose only two building blocks, the additive and the multiplicative structures, whose compositions will form schemes for two-step problems.

Schemes for solving one and two-step problems have been widely used in Israel during last fifteen years (Centre for Educational Technology et al. 1975) and motivated Shalin's Dissertation (Shalin and Bee, 1985). To illustrate such a scheme, let us first examine the scheme for Problem 3 (above). Problem 3 consists of a composition of two structures (additive and multiplicative). The connection between the two structures is formed so that the sum of the additive structure becomes one of the factors of the multiplicative structure. The general form of the scheme for Problem 3 will be as presented in Figure 1and its solution is achieved by calculating A:B-E.

(Figure 2 about here)

Let us now examine Problem 4 which has the same underlying structure and the same scheme as Problem 3. It actually describes the same situation, but differs in the designation of its complete and incomplete components.

There are 7 vases with 3 Tulips and 2 Roses in each vase.

How many flowers are there in all the vases? a significant factor in predicting levels of difficulty of word problems. If we restrict ourselves to two binary operation problems and consider all possible combinations of the four basic mathematical operations (addition, subtraction, multiplication and division) we arrive at 16 distinct types of problems. When non-commutative operations are involved, the order of the operations needs to be considered too, to distinguish for example, between a:(b+c) and (b+c):a ; or a:b+c (Note, that while all involve addition and division, they are different combinations of addition and division). The above three phrases denote a more basic distinction, i.e., which operation will be performed first, and which will be the second, thus, taking into account the order of the operations, we arrive at 24 distinct problems.

(Table 1, about here)

As can be noted, we arrive at 24 categories merely on the basis of the operations and their order. However, a lesson from the one-step problems tells us that categorization by operations does not fully explain the difficulty of certain problems. Research findings show that for the same operation ( + or - )some problems are easier and some harder depending on their scheme's category - whether they are "change" or "compare" problems.

To be consistent with our previous textual analysis our basic unit of analysis should be a complete structure (a 3-place relation) and not an operation. Thus, while Gray has four basic building blocks (the four operations ( +, -, x, : ) for his elementary two-step problems, we propose only two building blocks, the additive and the multiplicative structures, whose compositions will form schemes for two-step problems.

Schemes for solving one and two-step problems have been widely used in Israel during last fifteen years (Centre for Educational Technology et al. 1975) and motivated Shalin's Dissertation (Shalin and Bee, 1985). To illustrate such a scheme, let us first examine the scheme for Problem 3 (above). Problem 3 consists of a composition of two structures (additive and multiplicative). The connection between the two structures is formed so that the sum of the additive structure becomes one of the factors of the multiplicative structure. The general form of the scheme for Problem 3 will be as presented in Figure 1and its solution is achieved by calculating A:B-E.

(Figure 2 about here)

Let us now examine Problem 4 which has the same underlying structure and the same scheme as Problem 3. It actually describes the same situation, but differs in the designation of its complete and incomplete components.

There are 7 vases with 3 Tulips and 2 Roses in each vase.

How many flowers are there in all the vases?

The scheme of Problem 4 is illustrated in Figure 3. Note the location of the question mark.

(Figure 3 about here)

The calculation for the solution of Problem 4 is: (E+F)xB .

The only difference between Problems 3 and 4 is in what is given and in what is asked (what are the complete and the incomplete components). Even the latent component in both problems is the same and shared in both underlying structures. However, Problem 4 calls for addition and multiplication for its solution, while problem 3 calls for division and subtraction. Of course, the routes to the solution are different for problems 3 and 4. In Problem 3 we began the solution with the multiplicative structure, but in Problem 4 we start with the additive structure. Working with schemes makes it clear where to start (what is the first operation to be performed). One always starts from the structure which has two complete components and whose output will become part of the structure which initially had only one complete component.

In categorization according to operations Problems 3 and 4 will be in two distinct categories, yet in the analysis according to schemes they fall within the same scheme. It is not merely parsimony, which is also important, that we seek in moving from categorization by operations to categorization by schemes, but a consistent continuation of our approach to one-step problems in which the semantic structure and not the operation is the basic unit to be considered when examining cognitive processes involved in solving word problems.

Given that any scheme (for two-step problems) consists of two structures, there are in principle, only three ways by which the two structures can be connected. We will examine in detail the example of connecting an additive and a multiplicative structure, although combing two additive, or two multiplicative structures is also possible. If any additive structure consists of three components - the whole, and its two parts, and any multiplicative structure consists of the product (parallel to the whole) and its two factors (parallel to the parts), then, the universe of combinations produces the following three schemes (Shalin and Bee, 1985):

Scheme (1) The whole of one structure becomes a part of the other structure (Examples 3 and 4 above), hereinafter this scheme will be called The Hierarchical Scheme (H).

Scheme (2) The two structures share one whole, hereinafter - The Sharing Whole Scheme (S-W).

Scheme (3) The two structures share one part, hereinafter - The Sharing Part Scheme (S-P).

Figure 4 presents graphically the three possible compositions of two structures. We will use Shalin's graphical notation, since it is a well-known tree representation also used by others (Resseur (1990) and Thompson (1988)).

(Figure 4 about here)

As shown previously Problems 3 and 4 belong to the same scheme, (Scheme A). Two more problems belong to the same scheme, for example:

Problem 5:

A total of 35 flowers are distributed equally among 7 vases. Each vase has 2 roses and the rest are tulips. How many tulips are there in each vase?

And finally,

Problem 6: A total of 35 flowers are arranged in vases.

Each vase has 3 tulips and 2 roses. In how many vases are the flowers arranged?

( Figure 5 about here)

It is important to note that the components of Problems 3, 4, 5 and 6 are basically the same. They differ as to which are the complete and the incomplete components. The two latent components are also the same in all 4 problems, and the four problems contain all possibilities of moving the incomplete component among the four explicit components. Thus the four examples for scheme (A) exhaust all the theoretical possibilities. Note that the typicality of Scheme (A) is the type of connection between the structures, especially which latent components are shared by the two structures.

In the examples for the Hierachical Scheme, 3 to 6, the sum (the whole) of the additive structure became a factor in the multiplicative structure. But we can think of other problems with the same scheme in which the product of the multiplicative structure becomes one of the addends (the parts) in the additive structure, for example: Problem 7:

There are 28 flowers, 13 of them roses and the rest tulips.

The tulips are distributed equally among 5 vases.

How many tulips are there in each vase?

The structure of Problem 7 is as follows:

(Figure 6 about here)

Here, too, one could think of three more problems having the same scheme, but differing according to the place of the incomplete component, i.e., asking about the number of vases, the total number of roses, or the total number of flowers altogether. We leave the exact formulation of these problems to the reader.

Scheme B:

In Scheme B the sum (or the product) of one structure is the product (or sum) of the other structure. Following is an example of a Scheme B (Sharing Whole) problem:

There are 20 boys and 12 girls in the camp. They are divided into 4 equal groups. How many children are there in each group?

The components of Problem 8 are:

Additive structure: [B] There are 20 boys in the camp.

[C] There are 12 girls in the camp

[A] How many children are there in the camp?

Multiplicative Structure: [A] There are [?] children in the camp

[E] There are 4 equal groups.

[F] How many children are there in each group?

The graphical representation scheme for this problem will be:

(Figure 7 about here)

In Scheme B (S-W), as seen from our analysis, the latent component which is shared by both structures is the sum (whole) of the additive structure and the product of the multiplicative structure. This is also the connection between the two structures. Again, the situation described by Problem 8 can serve as a source for other three problems, all of which would be connected by the sum (product) but would vary according to the location of the incomplete component. We could ask about the number of the girls, the number of the boys or the number of the groups and each would form a distinct problem.

Scheme C :

In scheme C (S-P) the part (or the factor) of one structure is also a factor (part) of the other structure. For example Problem 9:

Problem 9:

In the party there were 20 children, 12 of which were boys.

The 40 flowers that were left from the party were distributed equally among the girls. How many flowers did each girl get?

The components Problem 9 are:

Additive Structure: [A] There were 20 children in the party.

[B] There were 12 boys in the party.

[C] How many girls were there in the party (latent)?

Multiplicative Structure: [E] There were [?] girls in the party (latent).

[D] There were 40 flowers in the party.

[F] How many flowers did each girl get?

The scheme for Problem 9 is as follows:

(Figure 8 about here)

Again, we can derive three other problems from the same situation, depending on the location of the incomplete component, whether it is about the number of flowers, the number of boys, or the number of children.

All the examples presented so far, were taken from a composition of one additive structure and one multiplicative structure. However, one could also construct problems which are combinations of two additive structures, or two multiplicative structures. These were not included in our present study.

To conclude: In the anlysis of schemes we have distinguished three basic schemes, each of which potentiallycalls for four different types of problems depending on the location of the incomplete component. If we distinguish in the Hierarchical Scheme (Scheme (A)), between the subordinate structure being an additive or a multiplicative structure, and between working with an intensive or extensive quantity as the latent component, we end up with 21 problems only for the combinations of additive and multiplicative structures (See Table 2).

(Table 2 about here)

Based on the above analysis it would beinteresting to find out whetherthere is any empirical support for categorization according to schemes in terms of clusters of problems with the same, or similar degree of difficulty. The research questions were formulated as follows:

(a) Which variables most account for the variance in difficulty in solving two-step problems?,

and,

(b) What is the order of difficulty among the various types of two-step problems.

In order to answer the above mentioned questions we conducted a study that included all 21 types of problems combining one additive structure with one multiplicative structure, manipulating the following variables:

(1) The Schemes - with three values - The Hierarchical Scheme; The Sharing Whole Scheme; and The Sharing Part Scheme.

(2) The Operations - with eight values - all the combinations of two binary operations, in which one is additive (addition or subtraction) and the other multiplicative (multiplication or division), taking into account which is performed first. Thus we have the following types of operation combinations: addition and then multiplication ( +, x ); or, multiplication and then addition ( x, + ), and in a similar manner, all the rest: ( -, x ); ( x, - ); ( +, \ ); ( \, + ); ( -, \ ); ( \ , - ). If we do not take the order into consideration we end up with only four types of combinations of operations: ( +, x); ( +, \ ); ( -, x ); and ( -, \ ).

3) Intensive or extensive quantity in multiplication - with two values - intensive and extensive. Research on multiplicative word problems revealed that in multiplication problems of the "mapping rule" type (Nesher, 1988) there is a need to distinguish between intensive and extensive quantities. Therefore in our sample of problems we maintainedthe distinction of whether the additive structure and the multiplicative structures are connected by an intensive or extensive quantity. Therefore, whenever appropriate we have provided two variants of the same problem, one employing an intensive quantity, and the other employing an extensive quantity for the latent component.

(4) The Developmental trend - with 4 values - for different grade levels: grades 3, 4, 5 and 6. This was done mainly to gain information for curricular purposes.

(5) Context - with 4 values. To obtain more general findings, we replicated the above 21 problems in 4 different contexts, thus arriving at 84 problems.The main research question

Based on the above analysis it would beinteresting to find out whetherthere is any empirical support for categorization according to schemes in terms of clusters of problems with the same, or similar degree of difficulty. The research questions were formulated as follows:

(a) Which variables most account for the variance in difficulty in solving two-step problems?,

and,

(b) What is the order of difficulty among the various types of two-step problems.

In order to answer the above mentioned questions we conducted a study that included all 21 types of problems combining one additive structure with one multiplicative structure, manipulating the following variables:

(1) The Schemes - with three values - The Hierarchical Scheme; The Sharing Whole Scheme; and The Sharing Part Scheme.

(2) The Operations - with eight values - all the combinations of two binary operations, in which one is additive (addition or subtraction) and the other multiplicative (multiplication or division), taking into account which is performed first. Thus we have the following types of operation combinations: addition and then multiplication ( +, x ); or, multiplication and then addition ( x, + ), and in a similar manner, all the rest: ( -, x ); ( x, - ); ( +, \ ); ( \, + ); ( -, \ ); ( \ , - ). If we do not take the order into consideration we end up with only four types of combinations of operations: ( +, x); ( +, \ ); ( -, x ); and ( -, \ ).

3) Intensive or extensive quantity in multiplication - with two values - intensive and extensive. Research on multiplicative word problems revealed that in multiplication problems of the "mapping rule" type (Nesher, 1988) there is a need to distinguish between intensive and extensive quantities. Therefore in our sample of problems we maintainedthe distinction of whether the additive structure and the multiplicative structures are connected by an intensive or extensive quantity. Therefore, whenever appropriate we have provided two variants of the same problem, one employing an intensive quantity, and the other employing an extensive quantity for the latent component.

(4) The Developmental trend - with 4 values - for different grade levels: grades 3, 4, 5 and 6. This was done mainly to gain information for curricular purposes.

(5) Context - with 4 values. To obtain more general findings, we replicated the above 21 problems in 4 different contexts, thus arriving at 84 problems.

The dependent variable was: The success percentage for each problem as a measure of its level of difficulty. Table 2 presents the 21 problem types used in the study, as defined by their scheme and operation.

1) The magnitude of the numbers appearing in the problems. All were less than 100 and easy to operate with. All calculations could be performed by heart.

2) The order of the text: The order of the string in the text reflected the sequence of the numbers in the order in which they should be used, and the question was always at the end of the text.

The 84 problems were divided into 8 different disjoint questionnaires, so as to give each child only 10 or 11 problems, without repeating the same context more than two or three times (in different problems). All 8 kinds of questionnaires were distributed randomly in each class (see Appendix A for examples of the different questionnaires). The questionnaires were distributed to the children in their regular homeroom by a team consisting of the homeroom teacher and an examiner. Time was unlimited and completion of the questaionnaire did not take more than 45 minutes.

The questionnaires were distributed to about 2000 childern drawn from six elementary schools in Israel, ultimately obtaining a complete set of data for 1824 children of grades 3, 4, 5 and 6, 456 children at each grade level. Taking into account that we had eight different questionnaires we ended up with 57 children answering each specific problem (specified by grade level, context, scheme, operations and whether it is an I (Intensive) or E (Extensive) quantity.

The raw data for the level of difficulty for of the 21 problems, by context and by class, is presented in Appendix B. We will discuss the effect of each variable in the following order: context, grade level and then, the effect of intensivity. Finally, we will discuss the relative impact of each of the major variables operation and scheme.

Four different contexts were selected for the study, each from a content area well known to the children: (a) Groups of children and their division into sub-groups. (b) Packing candy for a birthday. (c) Roses and tulips distributed in garden beds, and (d) Two kinds of bottles of drinks for a party. See Appendix A for the detailed questionaires.

Table 3 will present the percentage of correct answers ateach context for each grade level.

(Table 3 about here)

An analysis of variance for each of the grades yielded a non-significant effect of context for grades 3 and 4 (F=.735, N.S.; F=.816, N.S., respectively) and a very small effect for grades 5 and 6 (F=5.235, p=.001; F=3.015, p=.029, respectively). This significance was obtained due to the large sample (n=684 for each context at each grade). Since the effect of context was very small and the same order was maintained among different grades for the analyses reported later, the four contexts are pooled together.

As expected, we could observe a developmental trend in the ability to solve the same word problems, according to grade level. Yet in each grade level the order of difficulty for the 21 types of problems is the same. We can demonstrate this by the correlation between the grades concerning the order of difficulty (See Appendix B).

Table 4 will present the Spearman Correlation Coefficients between every two grades and the entire sample.

(Table 4 about here)

Thus, from the aspect of the order of difficulty among the 21 problems, the same difficulty trends appear at each grade. The variance among grades is due to the developmental level of the children, but the type of problems maintains the relative difficulty level. As mentioned before, administering the same problems at four grade levels provides us with the information regarding the grade level at which each type of problem should be introduced.

The developmental trend is shown clearly in Table 5 in which problems that were solved successfully by more than 70% of the children at each grade level, are marked with an *.

(Table 5 about here)

As can be seen from Table 5, a clear developmental trend exists. Some problems are easily and successfully solved even in the 3rd or 4th grades (e.g. Problem 2), while others are solved succesfully only later ( e.g. Problems 4, 9, 10, 17, 18, 19, 20). Interestingly enough, problems such as Problem 16, remain difficult even for 6-graders. We will return to the explanation of the possible sources of difficulty in certain problems when we discuss the effect of the schemes and the operations.

In light of the consistency among the grades regarding the order of difficulty among the 21 problems, we have pooled in further analyses the results of all grades employing covariance analysis when looking at other variables.

18 of the problems in the sample differ in the intensivity variable. We thus had 9 pairs of matched problems, in which one had an intensive quantity as its latent component, and its counterpart had an extensive quantity as its latent component. Table 6 will present the percentage of success on each of these pairs.

(Table 6 about here)

As can be seen from Table 6, problems in which the latent component is an intensive quantity are harder to solve. Only in two cases was the difference not significant.

Scheme was one of our major variables. Table 7 will present the percentage of success for each scheme, in each grade level:

(Table 7 about here)

A One Way Analysis of Variance for each grade by scheme yielded significant effects (F=51.9, p=0.000 for grade 3; F=73.6, p=0.000 for grade 4; F=80.7, p=0.000 for grade 5; F=51.1, p=0.000 for grade 6 ). The analysis for the entire population also yielded a significant effect (F=162.2, p=0.000). A test for range (Scheffe Procedure), presents three homogeneous subsets, in which each of the three schemes constitutes a subset by itself at the significant level of .05.

Thus, schemes constitute a significant variable affecting the level of difficulty of two-step word problems. The Hierarchical Scheme is the easiest, the Sharing Whole Scheme is next in difficulty, and finally the Sharing Part Scheme is the most difficult for all grade levels.

Operations is a variable traditionally used to describe the level of difficulty of arithmetic word problems. Table 8 presents the four possible combinations and the percentage of success in each combination:

(Table 8 about here)

One way analysis of variance by operations yielded a significant result (F=213.07, p=0.000). When also taking into consideration the order of the operations, we distinguish 8 values of this variable ( i.e., first adding and then multiplying is considered different from first multiplying and then adding). Table 9 presents the categorization of our 21 problems according to operations, and their percent of success:

(Table 9 about here)

A T-test for each pair of operations ( Examining the change of order for the pairs: I vs. II; III vs. IV; yielded a significant effect (T=5.83, p=0.000; and T=7.73, p=0.000, respectively). However, for the other two pairs (V vs. VI; and VII vs. VIII) the effect was not significant (T=-1.82, p=.069; T=1.86, p=.063, respectively). These findings are in general a replication of the Gray and Young study and are in accord with their findings, except for the difference between I and II in which Gray's results are in the opposite direction, but the difference is not statistically significant. This can be explained by examining the schemes that are activated in each of Gray's cells. Table 10 presents the schemes for each of Gray's operations.

(Table 10 about here)

As can be learned from Table 10, the difference in the difficulty in each pair of problems sharing the same operations but in a different order, can be explained by the underlying scheme of each order. Sharing the same operations does not mean sharing the same scheme and again we see that the schemes have the power of explaining difficulties faced by children in solving word problems. In cases with the same underlying scheme this difference is not significant.

The gist of the findings about operations is that combinations of commutative operations are the easiest (+ and x) while the combinations of the non-commutative operations (- and /) are the hardest, with the rest in between. The order of operations had only partial infulence and therefore in further analyses

we will pool the various orders.

Since both schemes and combinations of operations were significant factors, we were interested in the interaction among these two variables, with covariance for grade level. As can be seen from Table 11 there are empty cells in the scheme and operation matrix because some combinations of operations cannot, in principle, appear in certain schemes. Table 11 presents the percentage of success according to scheme and operation:

(Table 11 about here)

Since some cells are empty we were able to performa two-way analysis only in parts. A two-way analysis of schemes 1,2 & 3, and the combinations of operations +,/ and -,x, yielded significant main effects for schemes and operations (F=40.6, p=0.000; F=22.48, p=0.000, respectively) as well as a significant interaction effect (F=28.4, p=0.000). Similarly a two-way analysis of variance of the schemes 1 and 3, and operations +,/ , -,X , and -,/ also yielded significant main effects for schemes and operations (F=393,7, p=0.000; F=188.4, p=0.000, respectively), and a significant interaction effect (F=15.4, p=0.000).

Thus, both variables, schemes and operations, are effective variables influencing the degree of difficulty of each two-step problem. Moreover, there is interaction among them, so that feach scheme, one could sequence the problems according to operations and vice versa, and within each combination of operations (except for +,/) the problems can be ordered according to their scheme. The order of difficulty among schemes is generally as follows, from the easiest to the hardest: The Hierarchical Scheme, The Shared Whole Scheme and The Shared Part Scheme. As to the operations, the order of difficulty from the easiest to the hardest is (x, +), (/, +), (x, -), (/, -). And finally, in examining the interaction we observe that schemes 1 (H) and 3 (S-P) yield the same order of difficulty among the operations except for the operation (/, -) in which Scheme 3 (S-P) is harder than Scheme 1 (H). Similarly Scheme 2 (S-W), which appears only in conjunction with the operation ( +, /) and ( x, -) behaves differently than Schemes 1 (H) and 3 (S-P).

In summarizing the findings we claim that the two main variables affecting the level of difficulty of a given two-step problem are the scheme and the operations, in this order. Each of these variables and the interaction between them were significant. The impact of the kind of operations on the level of difficulty of word problems is well documented (Gray & Young, 1940; Jerman and Rees, 1972). The significant and innovative finding of our study is that the "scheme" variable appeared to be a central and even more influential than the "operations" variable. This supports the theoretical analysis of the role of schemes in solving word is in accord with the findings of Shalin & Bee (1985) who has studied another set of two-step problems.

The fact that the impact of the scheme is so significant is evidence for the cognitive processes hypothesized to exist in solving word problems. Schemes are said to assist the solver in interpreting diffuse situations in terms of a more structured scheme with the components of the problem having defined roles as arguments in the scheme. The solver can expect and comprehend the entire semantic set relations, on the basis of familiar schemes. He can then distinguish between the given and missing numerical information, and assign to each component in the text its appropriate role in the scheme. The main contribution of a scheme is that the solver, in employing a scheme, analyses the given text (which is supposed to be a description of the real world situation) with an eye on an hypothesized mathematical structure to be applied to that specific situation. Thus, the solver is reading the verbal text and operating directly on the text while attuned to the possible mathematical relations. From this point of view the scheme is a bridge between the verbal formulation of a problem and its mathematical structure.

As analysed in Section 2, several different problems, which call for different operations, can be deduced from a given situation in the world which is represented by a single scheme. Moreover, after assigning roles to the different components in a scheme for each of the various problems, the mathematical operations are deduced in a deterministic manner. Thus, we regard the concept of a scheme from which the operations are directly derived to be relatively more basic in terms of the cognitive process involved in solving word problems. It should be noted that it is impossible to decide, on the basis of two given operations to which scheme they belong, but it is possible to deduce the operations from the scheme. The fact that there are only three basic schemes for 21 combinations of two-step problems, also emphasizes its parsimony. Our empirical study substantiates our theoretical claims about the centrality of schemes in solving two-step problems. It is also worth mentioning that neither categorization by operations nor by schemes suffices to explain by itself the variance in performance. When considering operations we gloss over different schemes and by considering schemes we gloss over different operations. Only the interaction between these two variables can account for most of the variance in problem solving performance.

Context was not found to be a significant variable. As to grade level, which represents the developmental trend, it was found that the ability to solve difficult problems increases with age, but the relative inherent difficulty of each problem remains stable for the various grades and depends on the schemes and the operations involved in each problem. The question of whether the shared component is an intensive or extensive quantity, was also a significant variable and should be taken into account in instruction. The above findings have some bearing on instruction. First, the developmental part of our study will facilitate planning instruction according to grade level and sequencing correctly the domain of two-step problems. We can now determine the level of difficulty for any two-step problem as predicted by the combination of scheme, operations, and type of quantity and position it correctly in the instructional sequence.

Second, and this may be more important, it suggests the option of using these schemes more explicitly in the process of learning two-step or more complex problems. One can now take advantage of the parsimony of having only three schemes for two-step problems and make it known to the student. Simplifying the domain by reducing the vast amount of problems from different contexts or operations into three main schemes might enhance the use of proper cognitive processes in solving complicated problems. We believe that this approach can be extended to more complex problems merely by embedding more underlying structures in more complex schemes, but the fact that our building blocks are complete structures and schemes might enhance the solution process.