The role of schemes in designing computerized environments

Schemes are said to underly cognitive processes, including problem solving in Mathematics. An attempt to test the impact of schemes in solving 2-step word-problems was made in an instructional study where two computerized environments were compared. In the S.P.A. (Schemes for Problem Analysis) software, the user works with schemes in order to analyze and solve the problems. When using S.P.A., the solver has to fill in the data in a scheme which is presented graphically. From the scheme the operation is derived in a deterministic manner, and the user (solver) can then calculate the solution. A feedback mechanism follows. In the A.P. (Algebraic Proposer) software the user fills in a table with the data given in the word-problem and then has to find out which operation to use in order to find the solution. The effectiveness of S.P.A. vs. A.P. was empirically examined in an instructional experiment which studied two sixth grade classes during a four month period. The results of the experiment concluded that, in solving easy word problems, both of the two software systems were equally helpful. In harder word-problems, however, those who learned with the S.P.A. experienced more success than those who had learned with A.P. It was also found that S.P.A. was more instrumental in helping low achieving students cope with hard word-problems.     

Shifts in reasoning

The process of transition from a novice's state to that of an expert, in the constrained domain of decimals, is described in terms of explicit, intermediate, and transitional rules which are consistent, yet erroneous. These rules can be traced to former rules already established in earlier knowledge domains. Empirical data from children at grades 6, 7, 8 and 9 will demonstrate the evolution of an expert's knowledge through an elaborated learning path.     

Situation model, Text Base and what else? Factors affecting Problem Solving

Our aim in this paper is toidentify factors affecting problem solvingstrategies in the case of a multiplicativecomparative situation, involving threeunknown quantities whose sum is known, andin which comparison relations between twopairs of the quantities are given. Wepropose a model of the complexity of a wordproblem about such situation. The model hasseveral variables, namely, 1) the ratio ofthe number of quantities that are beingcompared to the number of referencequantities (the `reference ratio'); 2) thescheme of the situation, i.e. the type ofrelation between the two given elementarycomparison relations; 3) the order ofpresenting the elementary comparisonrelations; 4) the words used to describethe multiplicative comparison relations (somany times `more' Vs so many times `less').The choice of the values of these variablesin the formulation of a problem is assumedto affect the solver's solution strategy.The most important construct of the modelis what we have called the `complexitylevel' of a solution, namely, the number ofoperations (theoretically) needed to gofrom the text of the problem to analgebraic representation of this solution.An analysis, in terms of the model, of wordproblems that can be formulated about thiskind of situation makes it clear that,while there are several options in solvinga given problem in algebraic terms, some ofthem are simpler (have a lower level ofcomplexity), from the formal standpoint,than others. In fact, the most efficientchoice for the independent variable is onewhich, in the scheme of the situation,stands in the position of the `connector'between the two elementary comparisonrelations. The model is used in analyzingsolution strategies of a set of twelveproblems about this type of multiplicativecomparative situation by 104 teachers and132 15 years old students. In particular,we identify their choices of theindependent variable and discuss theirsolutions in terms of levels of complexity.We also look at their preferences forinterpreting the texts of the problems interms of the word `more' rather than`less'. One of our observations is that,even if they were not aware of it, thesubjects in the study preferred solutionswith a minimal route. There were, however,some exceptions in cases where subjectspreferred to use the least value as theindependent variable, as well as totranslate the `less' relations into the`more' relations regardless of the highcomplexity of this solution. This revised version was published online in August 2006 with corrections to the Cover Date.     

The role of schemes in two-step problems: Analysis and research findings

This study focuses on schemes involved in solving two-step word problems in which one of the two operations is either addition or subtraction and the other is either multiplication or division. Analysis of the problems yielded three possible basic compound schemes: (1) the Hierarchical Scheme, (2) the Shared-Whole Scheme, and (3) the Shared-Part Scheme. Twenty-one problems, in four different contexts, given to about 2000 students in Grades 3 to 6. It was found that Schemes, Operations and their interactions affect the difficulty level of the problems and that the order of difficulty of the 21 problems remained constant at the various grade levels.     

The development of semantic categories for addition and subtraction

Research conducted in several countries has shown consistent patterns of performance on change, combine and compare word problems involving addition and subtraction. This paper interprets these findings within a theoretical framework that emphasizes the development of empirical, logical and mathematical knowledge.