Pearla Nesher & Sara Hershkovitz ISRAEL   Most links are examples taken from our textbooks. In order to view them, you have to download the software Acrobat Reader (help).   Solving word problem as the exemplifications of various application is central to the series "One, Two, and…Three" (Nesher, 1989). Each year, starting from grade 1 (see appendix), a major chapter deals with solving word problems.   The heart of our program is the schematic approach to problem solving. This means that when teaching mathematical operations we keep in mind the schemes that will develop and serve the student later on in applying the formal language to various applications.   Our approach is based on the accumulated research of the past two decades on various types of word problems such as: 'combine', 'change' and 'compare' for the additive relation (Nesher, Greeno, & Riley, 1982b) ; and 'mapping rule', 'compare' and 'Cartesian product' for the multiplicative relation(Nesher, 1988; Vergnaud, 1983; Vergnaud, 1988), as well as their combinations in two (and more) step problems.

Additive Problems We take into consideration the developmental trends, and degree of difficulty determined by cognitive research. We hardly try to avoid introduction of misconceptions in the form of "keywords" (Nesher & Teubal, 1975), or other artificial crutches. We devote special care to building up schemes that represent the underlying structure of typical classes of problems. In so doing, we demonstrate that one should carefully attend to the text in the word problem, rather than to the numbers.   For example: If one attends only to the numbers and neglects the text, then some poor solvers are operate in the following manner: "Bla, bla, bla ….8, bla, bla, bla .2. How many……?" . In such a case one doesn't know what operation to perform (the poor solvers usually add…). The information is not encapsulated in the numbers, but rather in the text and the relations described by it.   For example: the following problem can be solved, even though no number is given: There are x boys and y girls in the classroom, How many children are there in the classroom?

In the following we describe in detail the flaw of learning: We begin introduction to the additive relation in grade 1. In our approach, the additive relation is a three-place relation that incorporates the addition and subtraction operations. We exemplify it by a three-rod-structure later to be generalized to three place additive number relation.   Before attending to word problems, the young child has to recognize additive relations among sets, presented to her by pictures. While working with sets the focus is on combining disjoint sets. The union set is defined by the actual objects of the subsets and not by their numbers. The child has to choose between the possibilities in which both sets of numbers work, but only one of these consists of three sets, two of which are disjoint subsets and the third is their union.   Knowing the connection between a union of sets and their additive structure (in numbers) we now present a verbal description of sets, and move on to word problems. Since we stress the relation among sets as expressed verbally (and not by the numbers) we now extend the additive structure from numbers to sets that include their numbers and their verbal descriptions.

This process serves us in various contexts such as: Combine, Change and Compare additive problems (Carpenter, Moser, & Romberg, 1982; De Corte, 1987; Nesher, Greeno, & Riley, 1982b; Nesher, 1978a; Vergnaud, 1982). We also take care to avoid reliance on verbal cues, or specific numbers. In order to recognize the additive scheme we also give the students problems with superfluous information or missing information, as well as open-ended problems that call for an additive scheme. We also address the generality of the additive relation beyond the specific numbers.   Multiplicative problems The multiplicative relation is exemplified by two means: Quisenaire Rods and an Array. As in the additive case we first present the multiplicative triple number relation (the factors and the product) without distinguishing between multiplication and division.   In passing to word problems the two factors are not interchangeable and therefore special care should be taken for the different types of multiplication word problems: 'Mapping Rule' (repeated addition); 'Multiplicative Compare' and the 'Cartesian Product'. We start with the 'Mapping Rule' problems that call attention to the intensive quantities (Schwartz, 1982 ) .We introduce the intensive quantity via the 4-place relation suggested by (Schwartz,1982 ; Vergnaud, 1983; Vergnaud, 1988). For example: 5 monkeys ate all the bananas. Each monkey ate 20 bananas. How many bananas were there?   Subsequently, after understanding the nature of the 'intensive quantities' we encapsulate it into the three place multiplicative relation:

As in the additive case, here too we present the students with a variety of contexts avoiding relying on superficial cues, and with open-ended problems.   More complicated problems (multi-step problems) Two-step (or more) word problems raise further difficulties. Unlike the one-step problem case in which all information is given explicitly in the text, in the case of more complex problems, the solver has to elaborate non-explicit items of information. For example: There are 7 tulips and 9 roses in each vase. How many flowers are there in 5 vases? This problem calls for two operations: It calls for both additive and multiplicative schemes. Such schemes consist of 6 arguments.

However, the text contains an explicit description of only 4 arguments, 3 of which are given with their numbers and the forth is asked about (the number is missing). The other two are to be elaborated in one of the schemes (the additive in this case) and used as a factor in the second scheme (the multiplicative scheme).   The second difficulty lies in the fact that it is difficult to comprehend and exhaust all the possible combinations of the four binary operations that comprise the two-step word problems. Employing the additive and multiplicative scheme enables to reduce dramatically the possible number of combinations, and identify the most general structure of two step problems, as follows: Having to map all the variety of two-step problems into three most general schemes enables the students to cope more systematically with complex problems.   The above analysis enabled us to categorize problems according to levels of difficulty as captured by the schemes. The schemes became a didactical tool for teaching the most difficult problems (Hershkovitz & Nesher, 1996; Hershkovitz & Nesher, 1997; Hershkovitz, Nesher, & Yerushalmy, 1990; Nesher & Hershkovitz, 1994), and it also helped organize this subject matter around the developmental trends. To promote this approach, a computerized software: "SPA" (Schemes for Problem Analysis), was developed and already tested as an helpful didactical tool.

The Sequence of Instruction We start by informing the children that part of the information would not be provided explicitly and that they will have to discover it themselves. We then introduce the Hierarchical Scheme and present variety of problems stemming from the situation. Next, we present the Shared-Whole Scheme in all its varieties. We emphasize the fact that the same situation can call for different questions and therefore lead to other mathematical operations.   Next, we introduce the Shared-Part Scheme in the same manner. We introduce the children to the different problems from the same scheme.   Finally, we end up with more complicated problems that use all the schemes, including open-end problems. We enrich the variety of problems by building a more natural context. In all cases we strive to help the child build up the most general schemes possible.

Bibliography Appendix