Why elevator trips are comparable to piano lessons and not to pet-sitting: world semantics guiding analogies between arithmetic word problems

Abstract : Arithmetic word problems of analogous objective mathematical structure can lead to dramatically different success rates depending on their wording (Hudson, 1983), and transfer between problems can be significantly favored or hindered by the cover stories used (Bassok, Wu & Olseth, 1995). But what promotes the perception of the analogy between different wordings of a same problem? Previous works have hinted at the existence of abstract semantic dimensions stemming from the solvers’ knowledge about the world and influencing the encoding and solving of problem statements (Bassok, Chase & Martin, 1998; Gamo, Sander & Richard, 2010; Gros, Sander & Thibaut, 2016). In order to assess the role of such world semantics on the solvers’ ability to perceive an analogy between problem statements, we created isomorphic problems that were implemented with different quantities. Half of these quantities were ontologically cardinal in that they described unordered elements that are usually grouped together (problems featuring, for example, turtles and iguanas being kept in a vivarium, or red and blue marbles being stored in a bag), and the other half were ontologically ordinal in that they featured values that are usually ordered along axes (problems involving piano lessons occurring over time, for example, or people using an elevator to go from one floor to another). Because solvers have experience with these quantities, we made the hypothesis that they would encode the problems according to their world semantics, abstracting a cardinal representation when cardinal quantities are used and an ordinal representation when ordinal quantities are used. The problems could all be solved using two distinct algorithms. Because they stress the grouping of unordered elements into parts and wholes, the cardinal representations naturally evoke a 3-step complementation algorithm consisting in calculating the values of each part so as to determine the values of the wholes. On the other hand, ordinal representations foster direct comparison of the wholes’ values without needing to calculate every part, promoting the use of a 1-step matching algorithm. Thus, the encoding of the problems into one of the two representations should result in the preferential activation of one of these two strategies. We tested this hypothesis by investigating solvers’ ability to perceive the structural analogy between the problems. We asked 191 participants (116 women, M=27.3 years, SD=11.3) to decide, among a list of 6 problems (3 ordinal problems and 3 cardinal problems) which ones could be solved similarly to a given source problem that was either ordinal or cardinal. Each participant was sequentially presented with 2 source problems (one ordinal and one cardinal) and 2 times 6 target problems. We predicted that the perception of analogy rate would be higher between two semantically congruent problems, that is two problems using cardinal quantities or two problems using ordinal quantities, rather than one of each. As hypothesized, analogy perception rate was higher between semantically congruent problems (83.8%) than between semantically incongruent ones (61.8%), F(1,187)=69.61, p<.001, ηp2=0.27. Additionally, a Wilk’s Lambda analysis for planned comparisons showed that for each individual quantity used, the analogy rate was significantly higher (479.54≤F≤1051.04; p<.001) when the source problem evoked a semantically congruent representation than when it evoked a semantically incongruent one. This result showed the dramatic influence of world semantics on the ability to detect similarities between problems. This result was then replicated in a second experiment presenting pairs of problems and asking for each if the problems could be solved similarly. This replication showed the robustness of the constraints imposed by world semantics on problem solving and confirmed the need for a model describing such interpretative effects in the encoding of arithmetic word problems.
Type de document :
Poster
Fourth International Conference on Analogical Reasoning, Jul 2017, Paris, France. 2017, 〈https://sites.google.com/site/analogy4paris/home〉
Liste complète des métadonnées

https://hal-univ-bourgogne.archives-ouvertes.fr/hal-01798952
Contributeur : Lead - Université de Bourgogne <>
Soumis le : jeudi 24 mai 2018 - 10:30:45
Dernière modification le : vendredi 8 juin 2018 - 14:50:07

Identifiants

  • HAL Id : hal-01798952, version 1

Collections

Citation

Hippolyte Gros, Emmanuel Sander, Jean-Pierre Thibaut. Why elevator trips are comparable to piano lessons and not to pet-sitting: world semantics guiding analogies between arithmetic word problems. Fourth International Conference on Analogical Reasoning, Jul 2017, Paris, France. 2017, 〈https://sites.google.com/site/analogy4paris/home〉. 〈hal-01798952〉

Partager

Métriques

Consultations de la notice

22