Through a glass darkly: A naturalistic study of students' understanding of mathematical word problems

A father is now 20 years older than his son. In 8 years, the father's age will be 5 years more than twice the son's age. Find their present age.Word problems, such as this one, are a perennial source of difficulty for students of school mathematics. Unfortunately, very little is known about why they are problematic, mainly because so little is known about how students understand such problems or the strategies they use in their efforts to solve them. Traditional research into word problems has shed precious little light on this question owing, in no small part, to its almost singular preoccupation with results of pupils' activities—as expressed in some sort of test score, and to its tendency to all but ignore what students actually do when confronted with problems of this kind.This study was carried out as one facet of a larger research project designed to gain more insight into some of the ways in which students understand school mathematics. It focuses on the efforts of one pupil, a twelve-year-old girl in grade seven, to come to terms with solving word problems using an algebraic approach. Strategies associated with both the structured and the unstructured clinical interview were used in order to reveal what was involved in her attempts to make sense of the word problems in her grade seven mathematics textbook.Based on the information gained in the interview, a rational reconstruction of the student's problem-solving strategy is proposed, and compared with the strategies normally prescribed in contemporary school mathematics textbooks. What emerges from this comparison is the finding that, while there appear to be systematic and fundamental differences between the procedures prescribed by the text and those actually used by the pupil in working through certain problems, these differences are undetectable in the finished product; either in the answer itself or in the rough or finished work. What this suggests, among other things, is that if, as educators and/or researchers, we limit our attempts to understand how students go about learning to solve word problems (or how they approach any other part of the school mathematics curriculum, for that matter) to examining what they commit to paper, we are apt to be seriously misled concerning what they genuinely understand and what they fail to understand. In short, if we are to learn more about why pupils experience difficulties with word problems we must begin to pay serious attention to what they say and do as they work their way through them.This study was supported by a Research and Development Grant from the Faculty of Education, Queen's University.