Abstract
This study investigated whether activating elements of prior knowledge can influence how problem solvers encode and solve simple mathematical equivalence problems (e.g., 3 + 4 + 5 = 3 + __). Past work has shown that such problems are difficult for elementary school students (McNeil and Alibali, 2000). One possible reason is that children's experiences in math classes may encourage them to think about equations in ways that are ultimately detrimental. Specifically, children learn a set of patterns that are potentially problematic (McNeil and Alibali, 2005a): the perceptual pattern that all equations follow an “operations = answer” format, the conceptual pattern that the equal sign means “calculate the total”, and the procedural pattern that the correct way to solve an equation is to perform all of the given operations on all of the given numbers. Upon viewing an equivalence problem, knowledge of these patterns may be reactivated, leading to incorrect problem solving. We hypothesized that these patterns may negatively affect problem solving by influencing what people encode about a problem. To test this hypothesis in children would require strengthening their misconceptions, and this could be detrimental to their mathematical development. Therefore, we tested this hypothesis in undergraduate participants. Participants completed either control tasks or tasks that activated their knowledge of the three patterns, and were then asked to reconstruct and solve a set of equivalence problems. Participants in the knowledge activation condition encoded the problems less well than control participants. They also made more errors in solving the problems, and their errors resembled the errors children make when solving equivalence problems. Moreover, encoding performance mediated the effect of knowledge activation on equivalence problem solving. Thus, one way in which experience may affect equivalence problem solving is by influencing what students encode about the equations.
Highlights
A crucial step in solving problems is noticing features that are relevant for their solution
We examined the effects of activating knowledge of the three operational patterns on undergraduate students’ encoding of mathematical equivalence problems
Were participants in the knowledge activation condition more likely to make reconstruction errors, they were more likely to encode the equivalence problems as if they were traditional arithmetic problems. These findings resemble the patterns found in children, in which knowledge of the operational patterns correlates with difficulties reconstructing equivalence problems (McNeil and Alibali, 2005a). It appears that one way in which early arithmetic experience may hinder performance on equivalence problems is by promoting inaccurate encoding, misencoding of such problems as traditional arithmetic problems
Summary
A crucial step in solving problems is noticing features that are relevant for their solution. In solving an arithmetic problem, it is necessary to note whether the operator symbol is a plus, minus, times, or division sign. Proper encoding of relevant problem features can mean the difference between solving a problem correctly and getting it wrong. Learners’ prior knowledge may provide them with information about what sorts of problem features are important. For example, numbers and operation symbols are important features, whereas other features (e.g., the font in which a problem is printed) can be safely ignored
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