Improving examples to improve transfer to novel problems

People often memorize a set of steps for solving problems when they study worked-out examples in domains such as math and physics without learning what domain-relevant subgoals or subtasks these steps achieve. As a result, they have trouble solving novel problems that contain the same structural elements but require different, lower-level steps. In three experiments, subjects who studied example solutions that emphasized a needed subgoal were more likely to solve novel problems that required a new approach for achieving this subgoal than were subjects who did not learn this subgoal. This result suggests that research aimed at determining the factors that influence subgoal learning may be valuable in improving transfer from examples to novel problems.     

Productive Failure in Learning Math

When learning a new math concept, should learners be first taught the concept and its associated procedures and then solve problems, or solve problems first even if it leads to failure and then be taught the concept and the procedures? Two randomized‐controlled studies found that both methods lead to high levels of procedural knowledge. However, students who engaged in problem solving before being taught demonstrated significantly greater conceptual understanding and ability to transfer to novel problems than those who were taught first. The second study further showed that when given an opportunity to learn from the failed problem‐solving attempts of their peers, students outperformed those who were taught first, but not those who engaged in problem solving first. Process findings showed that the number of student‐generated solutions significantly predicted learning outcomes. These results challenge the conventional practice of direct instruction to teach new math concepts and procedures, and propose the possibility of learning from one's own failed problem‐solving attempts or those of others before receiving instruction as alternatives for better math learning.     

Effects of Variation and Prior Knowledge on Abstract Concept Learning

Learning abstract concepts through concrete examples may promote learning at the cost of inhibiting transfer. The present study investigated one approach to solving this problem: systematically varying superficial features of the examples. Participants learned to solve problems involving a mathematical concept by studying either superficially similar or varied examples. In Experiment 1, less knowledgeable participants learned better from similar examples, while more knowledgeable participants learned better from varied examples. In Experiment 2, prior to learning how to solve the problems, some participants received a pretraining aimed at increasing attention to the structural relations underlying the target concept. These participants, like the more knowledgeable participants in Experiment 1, learned better from varied examples. Thus, the utility of varied examples depends on prior knowledge and, in particular, ability to attend to relevant structure. Increasing this ability can prepare learners to learn more effectively from varied examples.     

A process model for water jug problems

We develop and evaluate a model for the water jug task in which a subject is required to find a sequence of moves (pouring operations) which produce a specified amount of water in each jug. Experiment 1 was designed to evaluate the meansends, move selection heuristics that are assumed by the model. Experiment 2 tested the model's predictions concerning those aspects of the water jug task that determine problem difficulty.A three stage process model incorporating GPS-like, means-ends heuristics and assumptions concerning the utilization of short- and long-term memory was able to account for differences across problems as well as details of the performance of subjects solving a given problem. We conclude that a GPS-like model that only selects one move at a time (no forward planning of move sequences or setting up of subgoals) can provide a good account of solution behavior in the water jug task.     

样例的编码方式对问题解决迁移的影响及心理机制

以概率原理为材料,通过两个实验考察了样例的编码方式对问题解决迁移的影响及其心理机制。实验1探讨了样例的变异性与不同的自我解释模式结合对被试解决不同程度迁移问题的影响;实验2运用大声思维技术,进一步探讨不同样例的变异性和不同自我解释模式编码样例的条件下学习者的学习活动和问题解决的特点。结果表明:样例的变异性和诱发自我解释模式结合能有效地促进原理图式的获得和迁移;自我解释对理解原理和获得原理的概化图式有着重要作用。不同的学习者自我解释的特征并不一致,存在着个体差异。     

Evidence for abstract, schematic knowledge of three spatial diagram representations

Spatial diagram representations such as hierarchies, matrices, and networks are important tools for thinking. Our data suggest that college students possess abstract schemas for these representations that include at least rudimentary information about their applicability conditions. In Experiment 1, subjects were better able to select the appropriate spatial diagram representation for a problem when cued to use general category information in memory about those representations than when cued to use specific example problems given during the experiment. The results of Experiment 2 showed that the superior performance in the general category condition was not based on a comparison of the test problems with examples in memory. The results of Experiment 3 showed that the superior performance was not due to learning that occurred during the experiment or to transfer appropriate processing. The General Discussion section considers the nature of students' representation schemas and the question of why college students have only rudimentary schemas for common and widely applicable diagrammatic representations.     

类比比较样例对物理综合题学习迁移效果的影响

研究通过三个实验,比较了普通样例和类比比较样例在学习物理综合题中的效果,并在此基础上,进一步分析两种不同类型的类比比较样例对物理综合题学习的迁移效果以及“提示”这一样例设计方法的促进作用。结果发现:(1)通过类比比较样例学习物理综合题的效果优于普通样例;(2)相比于单综概念类比比较样例,综合概念类比比较样例能更好地提高物理综合题学习的近迁移成绩,但未提高其远迁移成绩;(3)提示能明显促进类比比较样例在物理综合题学习中的作用,但不同提示类型的影响无显著差异。该结果表明:类比比较样例是学习物理综合题的有效样例形式,而其中综合概念类比比较样例效果最佳;开放式提示和支架式提示均能较好地促进类比比较样例的学习。     

Learning to Solve Compare Word Problems: The Effect of Example Format and Generating Self-Explanations

A series of 3 experiments was conducted to examine factors that influence learning to solve 2-step arithmetic word problems by studying worked examples. Experiment 1 compared studying worked examples with conventional problem solving. Third graders presented with worked examples showed superior test performance to those required to solve conventional problems. Experiments 2 and 3 contrasted split-attention and integrated worked examples and investigated the influence of generating self-explanations. Significant split-attention effects were observed. Children presented with integrated worked examples outperformed those presented with split-source examples. Self-explanations further elucidated the distinction between integrated and split-source worked examples, but there was no significant effect of asking learners to generate self-explanations. Implications for word-problem-solving instruction are discussed.     

The influence of instructions on the adjustment of scales

In each of three experiments, one group of subjects first judged the eight largest and another group first judged the eight smallest of a graded series of 16 squares according to size. After four blocks of trials, both groups were shifted to the total series. The subjects changed their scales only slightly when no specific instructions were given at the point of shift (Experiment 1). They were able to maintain their old scales or construct new ones, if instructed to do so (Experiment 2). They were also able to do both things in parallel (Experiment 3). An additional finding was that responses to the "old" stimuli (those being part of the training series) were slower than responses to the "new" ones (those introduced after the shift), independently by of instructions. Furthermore, responses to the old stimuli were faster when these had to be rated on the new scale, and responses to the new stimuli were faster when these had to be rated on the old scale. The findings are interpreted as supporting a conditioning theory of judgment.     

Learning Strategies and Transfer in the Domain of Programming

We report two studies involving an intelligent tutoring system for Lisp (the Camegie Mellon University Lisp Tutor). In Experiment 1, we developed a model, based on production system theories of transfer and analogical problem solving, that accounts for effects of instructional examples, the transfer of cognitive skills across programming problems, and practice effects. In Experiment 2, we analyzed protocols collected from subjects as they processed instructional texts and examples before working with the Lisp Tutor and protocols collected after subjects solved each programming problem. The results suggest that the acquisition of cognitive skills is facilitated by high degrees of metacognition, which includes higher degrees of monitoring states of knowledge, more self-generated explanation goals and strategies, and greater attention to the instructional structure. Improvement in skill acquisition is also strongly related to the generation of explanations connecting the example material to the abstract terms introduced in the text, the generation of explanations that focus on the novel concepts, and spending more time in planning diminishing returns. Finally, reflection on problem solutions that focus on understanding the abstractions underlying programs or that focus on understanding how programs work seems to be related to improved learning.     

Effects of subliminal hints on insight problem solving

Two experiments tested a total of 509 participants on insight problems (the radiation problem and the nine-dot problem). Half of the participants were first exposed to a 1-min movie that included a subliminal hint. The hint raised the solution rate of people who did not recognize it. In addition, the way they solved the problem was affected by the hint. In Experiment 3, a novel technique was introduced to address some methodological concerns raised by Experiments 1 and 2. A total of 80 participants solved the 10-coin problem, and half of them were exposed to a subliminal hint. The hint facilitated solving the problem, and it shortened the solution time. Some implications of subliminal priming for research on and theorizing about insight problem solving are discussed.     

The testing effect and analogical problem-solving

Researchers generally agree that retrieval practice of previously learned material facilitates subsequent recall of same material, a phenomenon known as the testing effect. There is debate, however, about when such benefits transfer to related (though not identical) material. The current study examines the phenomenon of transfer in the domain of analogical problem-solving. In Experiments 1 and 2, learners were presented a source text describing a problem and solution to read which was subsequently either restudied or recalled. Following a short (Experiment 1) or long (Experiment 2) delay, learners were given a new target text and asked to solve a problem. The two texts shared a common structure such that the provided solution for the source text could be applied to solve the problem in the target text. In a combined analysis of both experiments, learners in the retrieval practice condition were more successful at solving the problem than those in the restudy condition. Experiment 3 explored the degree to which retrieval practice promotes cued versus spontaneous transfer by manipulating whether participants were provided with an explicit hint that the source and target texts were related. Results revealed no effect of retrieval practice.     

算术应用题二重变异样例学习的迁移效果

设计并运用了三种表面特征变异的二重样例和三种结构特征变异的二重样例,对二年级小学生解决算术应用题的迁移效果进行了实验研究。结果表明：一个原样例的学习就能明显促进了二年级小学生解决与原样例表面特征不同而结构特征相同的近迁移问题;学习三种表面特征变异的二重样例后没有产生明显的远迁移效果;而学习了三种结构特征变异的二重样例后产生了不同程度的远迁移效果,其中,规则平行组合二重样例学习的远迁移效果最好,其次是规则变异二重样例学习的远迁移效果,规则镶嵌组合变异二重样例学习的远迁移效果最差     

Conformity to Experimenter‐Provided Examples: Will People use an Unusual Feature?

Undergraduates studied nonword exemplars and then created their own novel nonwords. In Experiment 1, people studied legal or illegal nonwords. In Experiment 2, people studied illegal nonwords, but to increase awareness of the features, half of the participants assessed the features of the nonwords. Despite instructions to avoid copying any aspect of the examples, people incorporated features from the examples into their own novel creations. Although people do not spontaneously create illegal nonwords, those people that studied illegal nonwords were more likely to create illegal nonwords. Assessing the features of the examples did not reduce the likelihood that people would copy features of the examples. We discuss these results in the context of Ward's (1994) structured imagination account.     

Teaching a thinking strategy: Transfer across domains and self learning versus class-like setting

The purpose of this study is to investigate two issues related to the teaching of thinking skills: (1) transfer across domains; and (2) comparison between individual learning of a thinking strategy and ‘class-like’ setting, which consists of a didactic intervention that takes place in small groups. A microgenetic design was used, in which community college students engaged in investigation of problems (each student participated in 20 sessions). It was found that: (1) the control of variables strategy that was taught in the context of a problem in one domain transferred fully to a new problem in the same domain, hut transferred less well to an isomorphous problem in a different domain; (2) the ‘class-like’ setting contributed to improved performance (as measured by the frequency of valid inferences), with the didactic intervention seeming to contribute to this improvement; and (3) the learning environment described in this study had a larger effect on ‘slower learners’ than on ‘faster learners’. It is proposed to teach for enhanced transfer by focusing explicitly on recognition of underlying logical structures of content-rich contexts.     

Is case-based reasoning a source of knowledge generalisation?

This study examines the link between case-based reasoning and knowledge generalisation. Experiment 1 showed that case-based reasoning is not in itself a source of generalisation. In this experiment, subjects were able to adapt knowledge acquired during analysis of an example in order to successfully solve a very similar problem. But immediately afterwards, they failed on a problem symmetrical to the one they had just solved. Experiments 2A and 2B showed, in this same situation, that it is possible to lead subjects to generalise simply by stating that the practice problems were similar to the examples. These findings suggest that knowledge generalisation accompanying case-based reasoning is the result of a process that must be actively implemented by the subject, and not an automatic outcome of the source-to-target transfer.     

Transfer of conditional reasoning: Effects explanations and initial problem types

Transfer of reasoning on Wason's (1966) selection task was explored in three experiments. Experiment 1 tested the effects of problem explanations and verbalization instructions on transfer from abstract or thematic problems to abstract problems. Explanations facilitated transfer only when the initial problems were abstract; verbalization did not produce transfer between problems. Experiment 2 explored the effects of problem similarity and explanations on transfer between problems. Although transfer occurred following explanations, no effect of similarity was found for thematic problems. In both of these experiments, the thematic effect (Wason & Shapiro, 1971) was observed. Experiment 3 examined the effects of explanations to abstract or thematic problems on transfer to subsequent abstract or thematic problems. Transfer of reasoning occurred from both initial problem types, particularly to problems of the same type; however, transfer occurred to a greater extent from abstract problems than from thematic problems. The results are discussed in terms of problem similarity and Cheng and Holyoak's (1985) pragmatic reasoning schema hypothesis.     

Infant learning is influenced by local spurious generalizations

In previous work, 11‐month‐old infants were able to learn rules about the relation of the consonants in CVCV words from just four examples. The rules involved phonetic feature relations (same voicing or same place of articulation), and infants' learning was impeded when pairs of words allowed alternative possible generalizations (e.g. two words both contained the specific consonants p and t). Experiment 1 asked whether a small number of such spurious generalizations found in a randomly ordered list of 24 different words would also impede learning. It did – infants showed no sign of learning the rule. To ask whether it was the overall set of words or their order that prevented learning, Experiment 2 reordered the words to avoid local spurious generalizations. Infants showed robust learning. Infants thus appear to entertain spurious generalizations based on small, local subsets of stimuli. The results support a characterization of infants as incremental rather than batch learners.     

The inadvertent use of prior knowledge in a generative cognitive task

In four experiments with 332 participants, participants were asked to generate novel nonwords for English categories. When participants were shown examples embedded with regular orthographic structures, participants’ nonwords tended to conform orthographically to the examples, despite instructions to avoid using features of the examples. The effect was found with immediate testing (Experiment 1) and delayed testing (Experiment 2). The effect was also found with arbitrary features (Experiments 1–4), as well as with naturally occurring orthographic regularities (Experiment 4). Participants had difficulty avoiding the use of this prior knowledge, despite being able to list the features they were asked to avoid (Experiment 3). The results are discussed in terms of the inadvertent use of prior knowledge in generative cognitive tasks.