Contraries as an effective strategy in geometrical problem solving

Abstract

A focused review of the literature on reasoning suggests that mechanisms based upon contraries are of fundamental importance in various abilities. At the same time, the importance of contraries in the human perceptual experience of space has been recently demonstrated in experimental studies. Solving geometry problems represents an interesting case as both reasoning abilities and the manipulation of perceptual–figural aspects are involved.

In this study we focus on perceptual changes in geometrical problem solving processes in order to understand whether a mental manipulation in terms of opposites might help. Four conditions were studied, two of which concerned the search for contraries as an implicit or explicit strategy.

Results demonstrated that contraries, when used explicitly in solution processes, constitute an effective heuristic: The number of correct solutions increased, less time was needed to find a solution and participants were oriented towards the use of perception-based solutions—not only were perceptual solutions more frequent, but also, more specifically, the number of correct perceptual solutions increased. These last results concerning perception-based solutions were found both when participants were advised about the usefulness of the strategy and when they were not advised. Differences concerning which aspects of a problem were focused on during the solution process also emerged.

Keywords:

• Contraries in perception and reasoning
• Geometrical problem solving
• Contrast class
• Opposites

We would like to thank two anonymous reviewers and the Editor, Linden Ball, for their valuable comments.

No potential conflict of interest was reported by the authors.

Notes

¹ The biases related to remaining attached to the usual functions of everyday objects rather than seeing new possible uses of them (Duncker, 1926, 1935, 1945) and also imputing a procedure already repeated in a series of similar tasks to subsequent problems (known as the Einstellung effect by Maier, 1930, 1931a, 1931b, 1945; Luchins, 1942, 1946; Luchins & Luchins, 1950) are two variants of this.

² Representing numbers as points on a line is an example of representing abstract concepts in concrete terms (for a theory of embodied mathematics see Lakoff & Nunez, 2000).

³ This problem (1930) asks people to make four equilateral triangles out of six matches, with each triangle having one whole match for each side: Visualizing the matches arranged as a three-dimensional tetrahedron (each face corresponding to a triangle) significantly facilitates the task.

⁴ Some of us (Branchini, Burro, & Savardi, 2009) have analysed 10 classic problems used in research on problem solving to show that the solution processes often require the transformation of some of the initial properties of a problem into their contraries.

⁵ It is statistically appropriate to parameterise a multinomial model as a series of binomial contrasts (Alison, 1984; Dobson & Barnett, 2008).

⁶ Crawley (2012) suggested modelling data related to proportions in the glmer R function following a binomial distribution since proportions vary between 0 and 1.

⁷ But it is also a fundamental component of deductive reasoning in syllogisms (Evans, Handley, Harper, & Johnson-Laird, 1999; Johnson-Laird, 1983; Johnson-Laird & Bara, 1984; Johnson-Laird & Byrne, 1991).