Using Content to Interpret Structure: Effects on Analogical Transfer:

ion as "Sense Making" The studies described were de signed to show that when people attempt to understand novel prob lems, they operate under the as sumption that problem structures are compatible with semantic con straints implied by the problems' cover stories. In most cases, this assumption is justified, because the people who present students with word problems (e.g., teach ers, textbook writers) usually se lect cover stories that are compati ble with the appropriate problem structures. In fact, students of for mal domains are encouraged by teachers to adjust their abstract formal knowledge to the con straints implied by the entities that instantiate the variables in the learned equations (e.g., not to add distance to speed). Interestingly, it appears that the same students who achieve good understanding of the abstract formal concepts are those who are most likely to con strain their solutions by semantic knowledge implied by the prob lems' cover stories.8 Thus, unlike the case of content effects that re flect interference from associated content, when people use object attributes to constrain abstraction and transfer, their performance does not necessarily indicate poor understanding or insufficient ab straction from content. Rather, it might indicate their preference to reason and communicate at a level of abstraction that respects impor tant semantic distinctions. To demonstrate that interpre tive effects of content might reflect sensible rather than insufficient abstraction from content, Chase and I designed a task in which people's formal knowledge should make it easier for them to ignore rather than respect semantic con straints implied by object attrib utes. Specifically, instead of asking participants to solve unfamiliar word problems, we asked under graduate students to make up sim ple addition or division word problems for various pairs of ob ject sets (e.g., priests-ministers, priests-parishioners).9 The task in structions specified that the con Published by Cambridge University Press This content downloaded from 157.55.39.121 on Mon, 28 Nov 2016 04:01:58 UTC All use subject to http://about.jstor.org/terms CURRENT DIRECTIONS IN PSYCHOLOGICAL SCIENCE 57 structed problems had to involve the two object sets and the re quired arithmetic operation (either addition or division in a between participants design). Thus, partic ipants were free to choose the ac tual structure of the problems. The minimal solution that satis fies the requirements of the task is to construct problems in which the paired sets can be related directly by the required operation of addi tion (m + n) or division (m/n). This minimal solution was presented to participants in a sample problem that followed the task instructions. In principle, participants could construct such simple problems for every pair of object sets. How ever, our stimuli were designed so that the minimal solution was compatible with people's semantic knowledge for half of the paired object sets and incompatible for the other half. Specifically, the paired sets bore either a symmetric or an asymmetric semantic relation to each other (e.g., tulips-daffodils or tulips-vases, respectively). Symmetric pairs could be com bined to form meaningful super sets (e.g., tulips + daffodils = flowers), but addition of asymmet ric pairs would result in possible but unreasonable semantic ab stractions (e.g., tulips + vases = things). Similarly, asymmetric pairs could be meaningfully aligned with the operation and outcome of direct division (e.g., tulips/vases = tulips contained in each vase), but division of sym metric sets would result in possi ble but arbitrary proportional rela tions (e.g., tulips/daffodils = number of tulips per daffodil). That is, symmetric and asymmet ric object pairs were compatible, respectively, with direct addition and direct division, but not vice versa. Consistent with our interpretive view, participants were much more likely to construct simple problems for semantically compat ible than for semantically incom patible object pairs (81% vs. 59%, respectively). That is, despite par ticipants' overall preference to construct simple problems, in or der to achieve semantic compati bility they were ready to invest ex tra effort by constructing problems with a more complex mathematical structure. For example, when asked to construct a division prob lem for the symmetric pair tulips daffodils, participants would in vent another variable (e.g., p = days) and construct a division problem in which the paired sets were related by the compatible op eration of addition ([m + n]/p): "Wilma planted 250 tulips and 250 daffodils, and it took 20 days to plant them. How many did she plant per day?" The extra effort participants invested in construct ing semantically compatible com plex problems mirrors the inter pretive errors of participants who solved novel permutation prob lems.7 However, their perfor mance cannot be dismissed as in dicating insufficient abstraction from content.