SURFACE SIMILARITY CAN HIDE STRUCTURAL DIFFERENCES IN ALGEBRAIC PROBLEM-SOLVING BY TRANSFER

Transfer is usually defined as the ability to apply what is learned in one situation to a different new situation. Most of these learning situations are in fact ‘problematic situations’ referred to the real world or ‘problems’. The problematic situations are described in the problem statements. In order to solve these problems students have to connect the world situation with a scientific or mathematical implicit structure making inferences. In a traditional instruction practice teachers solve a set of example problems sharing some basic principle, law, theorem or procedure and later they propose ‘analogous’ problems to be solved by transfer. Transfer demands the construction of an analogy between the known situations and the new ones. The analogy between problems can be constructed in terms of their surface (real world features) or their structural (abstract, mathematical) similarity. Table I shows the nomenclature for problems having the same/different surface or structure

Same Structure & Same Surface: Equivalent / Same Structure & Different Surface: Isomorphic
Different Structure & Same Surface: Similar / Different Structure & Different Surface: Different
Table I: Relationship between problems in terms of their surface and structural similarity.

Reeves and Weisberg (1994) pointed out that perceptible features of the real-world objects and facts are easier to detect than abstract features, so the surface similarity between two situations promotes analogy construction easier than the structural similarity. Even, surface similarities can mask structural dissimilarities. In order to investigate this phenomenon in our context, a simple experiment was conducted. A group of 10th-grade students were asked to solve a set of 4 algebraic ‘target’ problems related to just one completely solved and explained ‘source’ problem. Each of the 4 target problems had a different relationship with the source problem in the terms shown in table 1, but the unknown variable was the same. We obtained two independent measures:
a)Perceived similarity between problems. Students evaluated in a five-point scale the help the source problem represented to solve every target problem.
b)Correct equations to solve the problem. For each target problem we offered 3 options (set of two equations) but only one correct. For the Similar and Different target problems one of the wrong options presented came from the source problem equations which were ‘copied and pasted’.
The main results obtained were:
1.-There were no significant differences in the perceived aid the source problem could provide to solve each of the four target problems. Students did not make differences due to the particular relationship between the solved example and the target problems.
2.-There was a significant lower level of success in the equations for the similar target problem. The most frequent error was to ‘copy and paste’ the equations from the solved example (negative transfer).
Thus, students did not perceive any particular difficulty in the activation phase (reading and comparing problems) in the similar problem but later, in the resolution phase, they made errors associated with negative transfer. These results replicate the findings by Reeves and Weisberg (1994) in the boundaries of this experiment.

References:
Reeves, L.M. & Weisberg, R.W. (1994). The role of content and abstract information in analogical transfer. Psychological Bulletin, 115, 381-400.