² IMPACT OF SEMANTIC ENCODING EFFECTS ON ARITHMETIC PROBLEMS IN SORTING AND SOLVING TASKS Bibliography • Gamo, S., Sander, E., & Richard, J.-F. (2010). Transfer of strategy use by semantic recoding in arithmetic problem solving. Learning and Instruction, 20, 400-410. • Martin, S. A., and Bassok, M. (2005). Effects of semantic cues on mathematical modeling: evidence from word-problem solving and equation construction tasks. Memory & Cognition 33, 471–478. Hippolyte Gros 1 , Emmanuel Sander 1 , & Jean-Pierre Thibaut 2 1 Paragraphe Lab, EA 349, University Paris 8, Department of Psychology, 2 Rue de la Liberté, 93526 Saint -Denis Cedex 02, France 2 LEAD-CNRS, UMR 5022, University of Burgundy, Pôle AAFE – Esplanade Erasme, 21065 Dijon. France INTRODUCTION 02 MATERIALS Semantic representations We believe that when facing a problem, the solver induces a semantic representation that differs from the mathematical structure, and depends on what he knows about the elements involved in the problem. For example, a problem involving apples and oranges will spontaneously evoke solving strategies adapted for additive problems, whereas a problem displaying vases and flowers will activate a representation compatible with solving strategies relying on multiplications and/or divisions (Martin & Bassok, 2005). The nature of the quantities In 2010, Gamo, Sander & Richard showed that the nature of the quantities involved in arithmetic problems could have a similar effect on semantic representations. They hinted that the use of quantities such as time units or distance units could lead to an ordinal representation of the values of a problem, whereas weight units would induce a cardinal representation instead. We tested this hypothesis using a sorting task and a solving task. 01 We created simple arithmetic problems that all shared the same formal mathematical structure (see figure 1). Values given: → Part 1, Whole 1, difference between Part 1 and Part 3 Question: → What is the value of Whole 2? Two solving strategies were always available : The 1-step matching strategy: • Whole1-Difference=Whole2. The 3-steps complementation strategy: • Whole1-Part1=Part2 • Part1-Difference=Part3 • Part2+Part3=Whole2. We hypothesized that only one strategy would be available at a time, depending on the semantic representation induced by the quantities used in the problem. Namely, problems involving time units, distance units or temperature units were believed to lead to an ordinal representation, congruent with the 1-step strategy, whereas problems with weight units, price units or number of elements evoked a cardinal representation, congruent with the 3-steps strategy. 03 SORTING EXPERIMENT 04 SOLVING EXPERIMENT 05 CONCLUSION → The difference we hypothesized between problems involving ordinal and cardinal quantities appeared both in the categories the subjects created, and in the solutions they provided to the problems. → Taken together, these results support the idea that the knowledge we have of the world deeply impacts the representation we create, and thus affects our ability to access the most relevant solving strategies. → This works expands the findings of Gamo & al. (2010) by showing that initial spontaneous encoding constrains the solving strategies even when the solution requires very simplistic operations such as small additions or subtractions. → In an educational perspective, we think that this is a promising finding highlighting the importance of semantic congruence mechanisms that need to be taken into account in order to allow for a semantic recoding of the problems. TASK 120 adults participated (M=27y9m, SD=9y8m). Subjects were asked to sort 12 problems depending on the strategies they would use to solve them. They were told that they could do as many categories as they wished. The problems only differed with regard to the nature of the quantities. TASK Participants : 60 adults participated (M=21y11m, SD=2y7m). Procedure : Subjects were instructed to solve 12 problems using as few operations as possible. There were: • Cardinal problems (problems with cardinal quantities) • Hybrid problems (problems with cardinal quantities but an ordinal context) • Ordinal problems (problems with ordinal quantities) RESULTS The results were consistent with our hypotheses : • The 1-step strategy ratio was more important for the ordinal problems than for the cardinal ones (t(56)=7,218, p<0,0001). • The hybrid problems led to a lower rate of 1-step strategy than the ordinal ones (t(56)=4,867, p<0,0001), and to a higher rate of 1-step strategy than the cardinal one (t(56)=2,217, p<0,05). ANALYSIS • Co-occurrence matrix → Study of the problems grouped together. • Proximity matrix → Perceived distance between the problems. • Multi-Dimensional Scaling → Projection of the distances on a 1-dimensional graph. MULTIDIMENSIONAL SCALING : FIRST DIMENSION KRUSKALL’S STRESS (1)=0,250 'Weight' problems 'Price' problems 'Number of elements' problems 'Duration' problems 'Distance' problems 'Temperature' problems -1,4 -1 -0,6 -0,2 0,2 0,6 1 Cardinal problems * 28,7% 37,7% 57,9% 71,3% 62,3% 42,1% 0% 20% 40% 60% 80% 100% Cardinal problems Hybrid problems Ordinal problems 1-step strategy 3-steps solving strategy * * PART 2 = COMMON PART WHOLE 2 PART 1 Cardinal: "Tom has 6 blue marbles." Ordinal: "Tom took piano lessons for 6 years." Cardinal: "Number of Lucy ’s blue marbles." Ordinal: "Duration of Lucy’s piano lessons." Cardinal: "Tom and Lucy have the same amount of red marbles." PART 3 DIFFERENCE WHOLE 1 ABSTRACT MATHEMATICAL STRUCTURE Ordinal: "Tom and Lucy started taking piano lessons at the same age." Cardinal: "Lucy has two red marbles less than Tom." Ordinal: "Lucy stopped her piano lessons 2 years before Tom." Cardinal: "Tom has 8 marbles in total." Ordinal: "Tom stopped his lessons at age 8. " Cardinal: "Total number of marbles Lucy has." Ordinal: "Age at which Lucy stopped taking piano lessons." FIGURE 1 The problems distribution on the main dimension of the model shows that spontaneous classifications did follow the Ordinal/Cardinal variation we introduced between the quantities. FIGURE 3 FIGURE 2 Contact • hippolyte.gros@gmail.com • emmanuel.sander@univ-paris8.fr • jean-pierre.thibaut@u-bourgogne.fr Ordinal problems Participants : Procedure: Financial support: GDRI-CNRS Acknowledgements