The interpretation of classically quantified sentences: a set-theoretic approach

We present a set-theoretic model of the mental representation of classically quantified sentences (All P are Q, Some P are Q, Some P are not Q, and No P are Q). We take inclusion, exclusion, and their negations to be primitive concepts. We show that although these sentences are known to have a diagrammatic expression (in the form of the Gergonne circles) that constitutes a semantic representation, these concepts can also be expressed syntactically in the form of algebraic formulas. We hypothesized that the quantified sentences have an abstract underlying representation common to the formulas and their associated sets of diagrams (models). We derived 9 predictions (3 semantic, 2 pragmatic, and 4 mixed) regarding people's assessment of how well each of the 5 diagrams expresses the meaning of each of the quantified sentences. We report the results from 3 experiments using Gergonne's (1817) circles or an adaptation of Leibniz (1903/1988) lines as external representations and show them to support the predictions.     

Apprendre à résoudre des problèmes en favorisant la construction d’une représentation alternative chez des élèves scolarisés en éducation prioritaire

In arithmetic problems solving, the representation of the problem spontaneously induced by the content of the statement is not always compatible with the solving strategy. This study evaluates a learning approach designed to lead pupils to build an alternative representation to the one spontaneously induced. In this paper, we evaluate the efficiency of this learning approach with children with learning difficulties enrolled in a priority education network. This learning by semantic recoding is a process based on an analogy between the problems and a comparison of strategies. The results show that after learning, the pupils built alternative representations.     

Learning new problem-solving strategies leads to changes in problem representation

Children sometimes solve problems incorrectly because they fail to represent key features of the problems. One potential source of improvements in children's problem representations is learning new problem-solving strategies. Ninety-one 3rd- and 4th-grade students solved mathematical equivalence problems (e.g., 3 + 4 + 6 = 3 + __) and completed a representation assessment in which they briefly viewed similar problems and either reconstructed each problem or identified it in a set of alternatives. Experimental groups then received a lesson about one or both of two solution strategies, the equalize strategy and the add–subtract strategy. A control group received no instruction. All children completed posttest assessments of representation and problem solving. Children taught the equalize strategy improved their problem representations more than those not taught it. This pattern did not hold for the add–subtract strategy. These results indicate that learning new strategies is one source of changes in problem representation. However, some strategies are more effective than others at promoting accurate problem representation.     

Computer-Based Learning: Graphical Integration of Whole and Sectional Neuroanatomy Improves Long-Term Retention

A study was conducted to test the hypothesis that instruction with graphically integrated representations of whole and sectional neuroanatomy is especially effective for learning to recognize neural structures in sectional imagery (such as magnetic resonance imaging [MRI]). Neuroanatomy was taught to two groups of participants using computer graphical models of the human brain. Both groups learned whole anatomy first with a three-dimensional model of the brain. One group then learned sectional anatomy using two-dimensional sectional representations, with the expectation that there would be transfer of learning from whole to sectional anatomy. The second group learned sectional anatomy by moving a virtual cutting plane through the three-dimensional model. In tests of long-term retention of sectional neuroanatomy, the group with graphically integrated representation recognized more neural structures that were known to be challenging to learn. This study demonstrates the use of graphical representation to facilitate a more elaborated (deeper) understanding of complex spatial relations.     

Representational Translation With Concrete Models in Organic Chemistry

In representation-rich domains such as organic chemistry, students must be facile and accurate when translating between different 2D representations, such as diagrams. We hypothesized that translating between organic chemistry diagrams would be more accurate when concrete models were used because difficult mental processes could be augmented by external actions on the models. In three studies, the task was to translate between different diagrams of molecules with or without a model. The model groups outperformed control groups, and students who received and used models outperformed those who received but did not use models. Uses of the model suggested that participants were performing external actions to support or replace difficult mental spatial processes. Spatial ability was a much weaker predictor of performance than model use. Results suggest that concrete molecular models can be an effective learning tool but some students need direct instruction to be able to take advantage of models.     

The psychological reality of practical representation

ABSTRACT

We represent the world in a variety of ways: through percepts, concepts, propositional attitudes, words, numerals, recordings, musical scores, photographs, diagrams, mimetic paintings, etc. Some of these representations are mental. It is customary for philosophers to distinguish two main kinds of mental representations: perceptual representation (e.g., vision, auditory, tactile) and conceptual representation. This essay presupposes a version of this dichotomy and explores the way in which a further kind of representation – procedural representation – represents. It is argued that, in some important respects, procedural representations represent differently from both purely conceptual representations and purely perceptual representations. Although procedural representations, just like conceptual and perceptual representations, involve modes of presentation, their modes of presentation are distinctively practical, in a sense which I will clarify. It is argued that an understanding of this sort of practical representation has important consequences for the debate on the nature of know-how.     

Skill transfer through goal-driven representation mapping

In this paper, we present an approach to transfer that involves analogical mapping of symbols across different domains. We relate this mechanism to Icarus, a theory of the human cognitive architecture. Our system can transfer skills across domains hypothesizing maps between representations, improving performance in novel domains. Unlike previous approaches to analogical transfer, our method uses an explanatory analysis that compares how well a new domain theory explains previous solutions under different mapping hypotheses. We present experimental evidence that the new mechanism improves transfer over Icarus’ basic learning processes. Moreover, we argue that the same features which distinguish Icarus from other architectures support representation mapping in a natural way and operate synergistically with it. These features enable our analogy system to translate a map among concepts into a map between skills, and to support transfer even if two domains are only partially analogous. We also discuss our system’s relation to other work on analogy and outline directions for future research.     

Beyond prototypes and classical definitions: Evidence for a theory-based representation of emotion concepts

The question of how people represent emotions is eminently important for a number of different domains of psychological research. The present study tested the assumption that emotion concepts are represented similar to theories in that they are comprised of a set of causally interrelated features. Using emotional scenarios and investigating the emotion concepts of anger, anxiety, and sadness it was found that people's representations of emotion concepts essentially involved the representation of the causal relation of emotion features and that the causal status of a feature determined the feature's centrality in the concept. Moreover, people's representations of emotion concepts corresponded to basic assumptions of cognitive emotion theories, namely that cognitive appraisals cause other surface features of emotion, like expression and behaviour. By providing evidence for a theory-based representation, the present results show that the classical and the prototype approach are insufficient to model the representation of emotion concepts.     

不同亚型数学学习困难儿童应用题问题表征过程的研究

采用自编数学应用题解决能力测验题对小学四年级单纯型数困儿童、混合型数困儿童和普通儿童问题理解阶段、问题整合阶段的差异,以及问题表征能力与数学问题解决之间的关系进行探究。结果表明：（1）单纯型与混合型数困儿童有效识别信息的能力弱,难于利用相关信息和排除干扰信息。（2）单纯型数困儿童比混合型数困儿童更擅于运用图式表征策略。（3）图式表征策略能促进数困儿童应用题的解决。     

The Use of Worked Examples as a Substitute for Problem Solving in Learning Algebra

The knowledge required to solve algebra manipulation problems and procedures designed to hasten knowledge acquisition were studied in a series of five experiments. It was hypothesized that, as occurs in other domains, algebra problem-solving skill requires a large number of schemas and that schema acquisition is retarded by conventional problem-solving search techniques. Experiment 1, using Year 9, Year 11, and university mathematics students, found that the more experienced students had a better cognitive representation of algebraic equations than less experienced students as measured by their ability to (a) recall equations, and (b) distinguish between perceptually similar equations on the basis of solution mode. Experiments 2 through 5 studied the use of worked examples as a means of facilitating the acquisition of knowledge needed for effective problem solving. It was found that not only did worked examples, as expected, require considerably less time to process than conventional problems, but that subsequent problems similar to the initial ones also were solved more rapidly. Furthermore, decreased solution time was accompanied by a decrease in the number of mathematical errors. Both of these findings were specific to problems identical in structure to the initial ones. It was concluded that for novice problem solvers, general algebra rules are reflected in only a limited number of schemas. Abstraction of general rules from schemas may occur only with considerable practice and exposure to a wider range of schemas.     

Mental representation and learning: The influence of practice on the development of mental representation structure in complex action

ObjectivesRecent research has elicited distinct differences in mental representations between athletes of different skill levels. Such differences suggest that the structure of mental representations changes as a function of skill level. However, research examining how such mental representation structures develop over the course of learning is lacking. In the present study, we examine the effects of practice on the development of one's mental representation of a complex action during early skill acquisition.DesignFor this purpose, we created a controllable learning situation, using a repeated-measures design with a control group. More specifically, novice golfers were randomly assigned to either a practice group (n = 12) or a control group (n = 12). Both groups were tested before and after an acquisition phase of three days as well as after a three day retention interval.MethodsMental representation structures of the putt were recorded, employing the structural dimensional analysis of mental representation (SDA-M), which provides psychometric data on the structure and grouping of action concepts in long-term memory. In addition, outcome performance of the practice group was measured, using two-dimensional error scores of the putt.ResultsFindings revealed a significant improvement in task performance, as well as functional changes in the structure of the practice group's mental representation. In contrast, no functional adaptations were evident in the mental representation of the control group.ConclusionOur findings suggest that motor skill acquisition is associated with functional adaptations of action-related knowledge in long-term memory.     

抽象概念表征的具身认知观

抽象概念是否通过感知经验来表征以及如何被感知经验表征是具身认知面临的一大问题.在抽象概念表征是否具有感知经验基础的问题上,具身认知理论认为抽象概念通过情境模拟或隐喻与感知经验发生联系.在抽象概念如何与感知经验表征发生联系的问题上,概念模拟理论强调情景或运动模拟在抽象概念表征中的直接作用;概念隐喻理论则侧重具体经验或具体经验与抽象概念之间的共同结构关系在抽象概念表征中的间接作用.未来研究应改变概念表征的稳定的心理实体观,从语言和抽象表征的关系、正常儿童和特殊群体的抽象概念表征差异入手,整合不同的具身认知观点.     

Mistakes on display: Incorrect examples refine equation solving and algebraic feature knowledge

Although findings from cognitive science have suggested learning benefits of confronting errors (Metcalfe, 2017), they are not often capitalized on in many mathematics classrooms (Tulis, 2013). The current study assessed the effects of examples focused on either common mathematical misconceptions and errors or correct concepts and procedures on algebraic feature knowledge and solving quadratic equations. Middle school algebra students (N = 206) were randomly assigned to four conditions. Two errorful conditions either displayed errors and asked students to explain or displayed correct solutions and primed students to reflect on potential errors by problem type. A correct example condition and problem-solving control group were also included. Studying and explaining common errors displayed in incorrect examples improved equation-solving ability. An aptitude-by-treatment interaction revealed that learners with limited understandings of algebraic features demonstrated greater benefits. Theoretical implications about using examples to promote learning from errors are considered in addition to suggestions for educational practice.     

In which direction to move? Facilitative and interference effects of gestures on problem solver's thinking

Recent research shows that co-speech gestures can influence gesturers’ thought. This line of research suggests that the influence of gestures is so strong, that it can wash out and reverse an effect of learning. We argue that these findings need a more robust and ecologically valid test, which we provide in this article. Our results support the claim that gestures not only reflect information in our mental representations, but can also influence gesturer's thought by adding action information to one's mental representation during problem solving (Tower of Hanoi). We show, however, that the effect of gestures on subsequent performance is not as strong as previously suggested. As opposed to what previous research indicates, gestures' facilitative effect through learning was not nullified by the potentially interfering effect on subsequent problem-solving performance of incompatible gestures. To conclude, using gestures during problem solving seems to provide more benefits than costs for task performance.     

Executive attention and task switching in category learning: Evidence for stimulus-dependent representation

One class of multiple-system models of category learning posits that within a single category-learning task people can learn to utilize different systems with different category representations to classify different stimuli. This is referred to as stimulus-dependent representation (SDR). The use of SDR implies that learners switch from subtask to subtask as trials demand. Thus, the use of SDR can be assessed via slowed response times, following a representation switch. Additionally, the use of SDR requires control of executive attention to keep inactive representations from interfering with the current response. Subjects were given a category learning task composed of one- and two-dimensional substructures. Control of executive attention was measured using a working memory capacity (WMC) task. Subjects most likely to be using SDR showed greater slowing of responses following a substructure switch and a greater correlation between learning performance and WMC. These results provide support for the principle of SDR in category learning and the reliance of SDR on executive attention.     

Language, perception, and the schematic representation of spatial relations

Schemas are abstract nonverbal representations that parsimoniously depict spatial relations. Despite their ubiquitous use in maps and diagrams, little is known about their neural instantiation. We sought to determine the extent to which schematic representations are neurally distinguished from language on the one hand, and from rich perceptual representations on the other. In patients with either left hemisphere damage or right hemisphere damage, a battery of matching tasks depicting categorical spatial relations was used to probe for the comprehension of basic spatial concepts across distinct representational formats (words, pictures, and schemas). Left hemisphere patients underperformed right hemisphere patients across all tasks. However, focused residual analyses using voxel-based lesion-symptom mapping (VLSM) suggest that (1) left hemisphere deficits in the representation of categorical spatial relations are difficult to distinguish from deficits in naming these relations and (2) the right hemisphere plays a special role in extracting schematic representations from richly textured pictures.     

Situation models,mental simulations,and abstract concepts in discourse comprehension

This article sets out to examine the role of symbolic and sensorimotor representations in discourse comprehension. It starts out with a review of the literature on situation models, showing how mental representations are constrained by linguistic and situational factors. These ideas are then extended to more explicitly include sensorimotor representations. Following Zwaan and Madden (2005), the author argues that sensorimotor and symbolic representations mutually constrain each other in discourse comprehension. These ideas are then developed further to propose two roles for abstract concepts in discourse comprehension. It is argued that they serve as pointers in memory, used (1) cataphorically to integrate upcoming information into a sensorimotor simulation, or (2) anaphorically integrate previously presented information into a sensorimotor simulation. In either case, the sensorimotor representation is a specific instantiation of the abstract concept.     

Developing procedural flexibility: Are novices prepared to learn from comparing procedures?

Background. A key learning outcome in problem‐solving domains is the development of procedural flexibility, where learners know multiple procedures and use them appropriately to solve a range of problems (e.g., Verschaffel, Luwel, Torbeyns, & Van Dooren, 2009 ). However, students often fail to become flexible problem solvers in mathematics. To support flexibility, teaching standards in many countries recommend that students be exposed to multiple procedures early in instruction and be encouraged to compare them. Aims. We experimentally evaluated this recommended instructional practice for supporting procedural flexibility during a classroom lesson, relative to two alternative conditions. The alternatives reflected the common instructional practice of delayed exposure to multiple procedures, either with or without comparison of procedures. Sample. Grade 8 students from two public schools (N= 198) were randomly assigned to condition. Students had not received prior instruction on multi‐step equation solving, which was the topic of our lessons. Method. Students learned about multi‐step equation solving under one of three conditions in math class for about 3 hr. They also completed a pre‐test, post‐test, and 1‐month‐retention test on their procedural knowledge, procedural flexibility, and conceptual knowledge of equation solving. Results. Novices who compared procedures immediately were more flexible problem solvers than those who did not, even on a 1‐month retention test. Although condition had limited direct impact on conceptual and procedural knowledge, greater flexibility was associated with greater knowledge of both types. Conclusions. Comparing procedures can support flexibility in novices and early introduction to multiple procedures may be one important reason.     

Who Benefits from Diagrams and Illustrations in Math Problems? Ability and Attitudes Matter

How do diagrams and illustrations affect mathematical problem solving? Past research suggests that diagrams should promote correct performance. However, illustrations may provide a supportive context for problem solving, or they may distract students with seductive details. Moreover, effects may not be uniform across student subgroups. This study assessed the effects of diagrams and illustrations on undergraduates' trigonometry problem solving. We used a 2 (Diagram Presence) × 2 (Illustration Presence) within‐subjects design, and our analysis considered students' mathematics ability and attitudes towards mathematics. Participants solved problems more accurately when they included diagrams. This effect was stronger for students who had more positive mathematics attitudes, especially when there was an illustration present. Illustrations were beneficial for students with high mathematics ability but detrimental for students with lower ability. Considering individual differences in ability and attitude is essential for understanding the effects of different types of visual representations on problem solving. Copyright © 2017 John Wiley & Sons, Ltd.