Despite word problems are important for mathematics curricula, students often experience difficulties with addition and subtraction word problem solving. Different hypotheses have been advanced to explain why children succeed or fail in this task. Prominent among these is the hypothesis that conceptual knowledge is an important foundation for successful problem solving. The present study has been designed to analyse this hypothesis.

 

How children solve simple addition and subtraction word problems has been of interest for a very long time. Classification schemes have been developed to categorize problems along the relevant dimensions, and, indeed, performance varies dramatically across the different categories of problems. Three classes of problem situations modelled by addition and subtraction can be distinguished. These are situations involving a change from an initial state to a final state through the application of a transformation (change problems), situations involving the combination of two discrete sets or splitting of one set into two discrete sets (combine problems), and situations involving the quantified comparison of two discrete sets of objects (compare problems). Within each of these three major semantic categories, further distinctions were made resulting in 14 different types of one-step addition and subtraction problems.

 

Several models have been developed that simulate young children’s understanding and solution of word problems that concern the exchange, combination and comparison of sets. Underlying these computer models was the general assumption that a skilful solution process of a word problem starts from a network representation of the basic semantic relationships between the main quantities in the problem, in terms of one of the three above-mentioned basic semantic structures. This network is considered the result of a complex interaction between bottom-up and top-down analysis; that is, the processing of the verbal input as well as the activity of semantic schemata contributes to the construction of this network representation. More difficult problem types require re-representations in terms of other schemata before a proper arithmetic action or operation can be selected and performed. For instance, according to the Riley et al. model, change problems with an unknown initial set or compare problems with an unknown reference set, can only be solved after the original problem representation in terms of, respectively, a change or a compare schema has been re-represented in terms of a part-whole structure. The more competent the problem solver, the more able (s)he is to process the text in a top-down way by relying on his(her) well-developed schemata. In this context, when above we mentioned the hypothesis that conceptual knowledge is an important foundation for successful problem solving, conceptual knowledge is to refer to understanding of the semantic relations (schemata) described by the text, this is, knowledge about increases, decreases, combinations, and comparisons involving sets of objects. In turn, to be able to operate with these semantic relations depends on acquisition of knowledge concerning part-whole relations, especially for complex problems. That is, semantic schemata identify which problem quantities are parts and which are wholes (mathematical knowledge in the words of Riley, Greeno and Heller, 1983). Other authors (e.g. Kintsch and Greeno, 1985; Cummins, Kintsch, Weimer and Reusser, 1988, Reusser, 1988) presented a more complex model in which text understanding is interwoven with the construction of a mathematical problem representation in terms of sets and their interrelations. These models suggest that difficulties with word problems arise from a lack of textual understanding (rather than with access to mathematical knowledge), which prevents children from making contact with relevant mathematical knowledge.

 

According to these ideas, an instructional program has been developed by us in order to improve the word problem solving ability of children with low-ability. The essential elements of this program are:

1. Textual processing (construction of the text base in terms of reading comprehension); children in their’s own words had to paraphrase or to re-estate problems problem information bringing together what they already know and what they don’t know yet from problem text.

2. Schema identification; this component is taught using schemata diagrams for each of the three problems types (change, combine and compare) and by providing the learner with strategies for mapping information from the verbal text onto elements of the schema.

3. Re-representation in terms of a part-whole structure; that is, to interpret the information represented in the problem by identifying the total or the larger quantity in a problem.

 

But our aim was not to show the effectiveness of the program, because it has been widely shown by other studies. Our goal was to analyze how different aids were incorporated by children. For this, we base our study on the resistance to the instruction concept, which involves an analysis of how the children take advantage of the external help provided by someone more expert (instructor) when solving problems. Thus, the instructors worked individually with a sample of children with difficulties in solving problems, providing them with help related to the different components of the program. In this way, the quantity of help that the children needed was recorded.