Content area
Abstract
Mathematics students exhibit prevalent, systematic, and robust errors in mathematical computations and problem solving. Studies have revealed errors systematic for particular individuals and across populations. These studies have relied on written work and documented and classified the errors. However, to understand and remediate these errors, we must discover what students understand about particular subject matter and what they are thinking as they make these errors. The goal of this study is to explore what students find difficult about algebra in order to illuminate the causes of their misconceptions.
This study examines the problem solving behavior of nine students in second year algebra. Each student was observed over a range of topics in algebra in order to reveal common faults in problem solving strategies rather than difficulty with particular subject matter. The format of the study was problem solving sessions which consisted of helping the students with their homework assignments, reviewing tests and quizzes, and answering the students' questions about mathematics. The students were prompted to use analytical techniques described by Polya by being asked the questions that an effective problem solver would address while solving a problem.
The most surprising result from this study was that lack of expertise with mathematical content was a more frequent impediment to solving algebra problems than problem solving deficiencies. Most students participating in this study had adequate problem solving skills but these skills were masked by inadequate basic skills; in fact, many of their errors resulted from efforts to continue the problem solving process. The deficiencies in subject matter which were most detrimental to problem solving were signed numbers, fractions, and the peculiar qualities of zero.
These students seemed to have stored structures both appropriate and inappropriate to the current problem. Their errors were typically the result of performing a mathematically valid operation which was not appropriate in the current context, but would be appropriate elsewhere. The common causes of this were misjudging the similarity of the current problem to other problems, focusing on an aspect of a problem irrelevant to selecting a strategy for solving the problem, and failing to test hypotheses.
