PEARLA NESHER
1
, SARA HERSHKOVITZ
2
and JARMILA NOVOTNA
3
SITUATION MODEL, TEXT BASE AND WHAT ELSE?
FACTORS AFFECTING PROBLEM SOLVING
ABSTRACT. Our aim in this paper is to identify factors affecting problem solving strate-
gies in the case of a multiplicative comparative situation, involving three unknown quant-
ities whose sum is known, and in which comparison relations between two pairs of the
quantities are given. We propose a model of the complexity of a word problem about such
situation. The model has several variables, namely, 1) the ratio of the number of quantities
that are being compared to the number of reference quantities (the ‘reference ratio’); 2)
the scheme of the situation, i.e. the type of relation between the two given elementary
comparison relations; 3) the order of presenting the elementary comparison relations; 4)
the words used to describe the multiplicative comparison relations (so many times ‘more’
Vs so many times ‘less’). The choice of the values of these variables in the formulation of a
problem is assumed to affect the solver’s solution strategy. The most important construct of
the model is what we have called the ‘complexity level’ of a solution, namely, the number
of operations (theoretically) needed to go from the text of the problem to an algebraic
representation of this solution. An analysis, in terms of the model, of word problems that
can be formulated about this kind of situation makes it clear that, while there are several
options in solving a given problem in algebraic terms, some of them are simpler (have a
lower level of complexity), from the formal standpoint, than others. In fact, the most effi-
cient choice for the independent variable is one which, in the scheme of the situation, stands
in the position of the ‘connector’ between the two elementary comparison relations. The
model is used in analyzing solution strategies of a set of twelve problems about this type
of multiplicative comparative situation by 167 teachers and 132 15 years old students. In
particular, we identify their choices of the independent variable and discuss their solutions
in terms of levels of complexity. We also look at their preferences for interpreting the texts
of the problems in terms of the word ‘more’ rather than ‘less’. One of our observations is
that, even if they were not aware of it, the subjects in the study preferred solutions with a
minimal route. There were, however, some exceptions in cases where subjects preferred to
use the least value as the independent variable, as well as to translate the ‘less’ relations
into the ‘more’ relations regardless of the high complexity of this solution.
KEY WORDS: comparison relation, complexity, multiplicative comparison, more – less,
reference, scheme, text-base, two-step word problems
1. I NTRODUCTION
Solving word problems in mathematics often means deducing new inform-
ation (mostly quantitative) from the given data. The known information
can be collected by the solver him or herself, or it can be provided by
Educational Studies in Mathematics 52: 151–176, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.