The representations of the arithmetic operations include functional relationships

Current theories of mathematical problem solving propose that people select a mathematical operation as the solution to a problem on the basis of a structure mapping between their problem representation and the representation of the mathematical operations. The structure-mapping hypothesis requires that the problem and the mathematical representations contain analogous relations. Past research has demonstrated that the problem representation consists of functional relationships, or principles. The present study tested whether people represent analogous principles for each arithmetic operation (i.e., addition, subtraction, multiplication, and division). For each operation, college (Experiments 1 and 2) and 8th grade (Experiment 2) participants were asked to rate the degree to which a set of completed problems was a good attempt at the operation. The pattern of presented answers either violated one of four principles or did not violate any principles. The distance of the presented answers from the correct answers was independently manipulated. Consistent with the hypothesis that people represent the principles, (1) violations of the principles were rated as poorer attempts at the operation, (2) operations that are learned first (e.g., addition) had more extensive principle representations than did operations learned later (multiplication), and (3) principles that are more frequently in evidence developed more quickly. In Experiment 3, college participants rated the degree to which statements were indicative of each operation. The statements were either consistent or inconsistent with one of two principles. The participants' ratings showed that operations with longer developmental histories had strong principle representations. The implications for a structure-mapping approach to mathematical problem solving are discussed.     

A dual-process account of digit invariance learning

Performance in the McGeorge and Burton (1990) digit invariance task was originally thought to be mediated by unconscious abstraction of a “rule” that identified the invariant feature across all study items. Subsequent explanations have suggested explicit strategy use or similarity-to-exemplar matching rather than abstraction. This paper presents data that suggest that both similarity and abstraction can be used under different task demands. Delay between study and test afforded abstraction of the invariant knowledge whereas reducing the pool of study exemplars enhanced responding based on specific similarity. These results parallel effects found in the categorization literature. Rule abstraction in this sense may be due to statistical learning of feature frequency rather than abstraction of a central tendency or a complex/conceptual rule. Categorizing responses into subjective memory states (remember/know/guess) demonstrates that neither the similarity matching nor the abstraction mechanism uses information from episodic memory. Confidence measures show that participants are more confident of responses when the prototypical representation is used but not specific similarity. Taken together, these data suggest that abstracted knowledge is not held consciously but that participants have meta-awareness of when they are using the abstracted representation.     

A dual-process account of digit invariance learning

Performance in the McGeorge and Burton (1990) digit invariance task was originally thought to be mediated by unconscious abstraction of a “rule” that identified the invariant feature across all study items. Subsequent explanations have suggested explicit strategy use or similarity-to-exemplar matching rather than abstraction. This paper presents data that suggest that both similarity and abstraction can be used under different task demands. Delay between study and test afforded abstraction of the invariant knowledge whereas reducing the pool of study exemplars enhanced responding based on specific similarity. These results parallel effects found in the categorization literature. Rule abstraction in this sense may be due to statistical learning of feature frequency rather than abstraction of a central tendency or a complex/conceptual rule. Categorizing responses into subjective memory states (remember/know/guess) demonstrates that neither the similarity matching nor the abstraction mechanism uses information from episodic memory. Confidence measures show that participants are more confident of responses when the prototypical representation is used but not specific similarity. Taken together, these data suggest that abstracted knowledge is not held consciously but that participants have meta-awareness of when they are using the abstracted representation.     

Comprehension of arithmetic word problems by fifth‐grade pupils: Representations and selection of information

Background: Understanding word problems leads to the construction of different levels of representation. Some levels specify the elements which are indispensable for solving the problem (problem model, PM) and others specify the agents, actions and events in everyday concepts (situation model, SM). Aims: By studying how the information is selected, we try to specify the nature of the representations constructed during the reading of a word problem: understanding a problem leads to the construction of two complementary levels of representation (PM and SM) or to the construction of only one representation (PM)? Sample: Ninety‐one fifth‐grade pupils (mean age 10 years 9 months) took part in this study and were divided into two groups according to their mathematical ability. Method: As well as the information considered as indispensable for solving the problems (solving information), different types of information (situational information) were introduced into standard word problems. In a first task, participants were asked to select the information in order to ‘make the word problem as short as possible’ (locate the elements used for developing PM). In a second task, they were asked to select the information in order to ‘make the word problem easier to understand’ (determine whether the participants developed a SM). Results: The participants successfully differentiated between the solving information and the situational information. An interaction was also observed between the type of information and the task. The mathematical ability of the participants was seen to have an influence on the selection of situational information. Conclusion: Understanding leads to the construction of two complementary representation levels: the problem model and the situation model.     

The memory representation of motor skills: a test of schema theory

Currently, a popular model for the central representation of motor skills is embodied in Schmidt's schema theory of discrete motor skill learning (Schmidt, 1975). Two experiments are reported here that contrast predictions from a schema abstraction model that is the basis for schema theory with those from an exemplar-based model of motor skill memory representation. In both experiments, subjects performed 300 trials per day of three variations of a three-segment timing task over 4 days of acquisition. The subjects then either immediately transferred to four novel variations of the same task (Experiment 1) that varied in degree of similarity to the exemplars experienced during acquisition; or performed two novel and two previously produced exemplars, following 24-h and 1-week retention intervals (Experiment 2). The results indicated that novel task transfer was not affected by the degree of similarity between the acquisition and transfer exemplars, and that there was no advantage for a previously produced exemplar over a novel exemplar after either a 24-hr or 1 week retention interval. Also, in both experiments, a consistent pattern of bias in responding was noted for novel task transfer and retention. These results are indicative of a schema abstraction model of memory representation for motor skills.     

青少年聋生的分类学联系

了考察青少年聋生分类学联系的特点,使用不同类别的基本水平概念,完成了语义归类和错误再认２个实验,并同健听生的结果进行比较。结果表明：⑴聋生和健听生有类似的类别意识,但聋生的类别意识较健听生为弱。⑵与健听生比,呈现方式对聋生的反应影响较小。聋生的分类学概念有较强的形象化倾向。青少年聋生分类学联系的特点与他们语言能力发展迟缓、使用自然手语有关。因此,应加强聋生的规范手语教学,重视培养聋生的抽象思维能力     

Unfixing Design Fixation: From Cause to Computer Simulation

This paper argues that design fixation, in part, entails fixation at the level of meta‐representation, the representation of the relation between a representation and its reference. In this paper, we present a mathematical model that mimics the idea of how fixation can occur at the meta‐representation level. In this model, new abstract concepts derived from the meta‐representation are not simple combinations of the design ideas expressed in the primary representation. Instead, the process of meta‐representation corresponds to the abstraction of new features at a higher, hierarchically separated level of representation from an existing one. We show a computational process of pattern extraction from existing design representations that results in the crystallization of design ideas or ‘chunks’ at an abstract level. The relation between the representation and its referent is explicitly described in these ‘chunks’, yet the ‘chunks’ have different properties from the properties of the original representation.     

数学解题过程的眼动研究

数学解题过程的眼动研究可划分为3个阶段：第一阶段是对数字运算过程的眼动研究,第二阶段是对数学应用题解题过程的眼动研究,第三阶段是对几何题解题过程的眼动研究。在我国,对数学解题过程的眼动研究尚为空白。加强这一领域的研究,可以深化对数学问题表征的认识,对数学学科的教与学具有重要意义。     

元认知在画图表征策略和小学生数学问题解决能力中的中介作用

该研究采用自然实验和问卷调查法,对268名小学五年级学生进行画图表征策略训练,在此基础上根据策略学习情况筛选出掌握了该策略的218名学生,以这些学生为被试,探讨元认知在画图表征策略和小学生数学问题解决能力中的作用。结果表明:(1)策略学习后,小学生的画图表征策略水平和数学问题解决成绩均显著高于策略学习前。(2)画图表征策略通过元认知这一中介变量对小学生数学问题解决能力有显著的促进作用。     

Problem solving with a simple transformation problem with and without continuous external memory support

Problem solvers can use external memory aids to represent problem information. In this study, an external problem representation was either continuously available or available on demand of the participants. Two experiments explore the hypothesis that despite the higher working memory load without a continuous external memory support, problem solvers try to reduce the cognitive costs by implementing deliberate problem solving strategies. Problem solving with and without continuous external memory support was found to be largely equivalent with high and low operator implementation costs and in problem spaces of different sizes.     

The impact of external representation in a rule discovery task

The 2-4-6 rule discovery task introduced by Wason (1960) elicits generally poor hypothesis-testing behaviour as reflected by the underwhelming effort and creativity participants invest in the process before announcing their “best” guess. The traditional task departs from real-world hypothesis testing in a significant respect: Reasoners proceed to discover the rule solely from an internal representation of the problem space. Two studies reported here examined representational effects in the 2-4-6 task by developing task isomorphs that offered external, physically manipulable, representations of the dimensions of the problem space. These isomorphs led to a significantly higher incidence of successful rule discovery, driven by significantly greater diligence and creative exploration of the problem space, compared to other task isomorphs that did not offer such an external representation. These results confirm the fruitfulness of a research programme that explores the contextual determinants of sound hypothesis testing.     

Cue abstraction and ideal prototype abstraction in estimation tasks

This study investigates abstracted processes and introduces a new prototype abstraction model adapted to estimation tasks. This prototype abstraction model assumes that the processing of whole exemplar patterns supports the detection of the underlying statistics necessary for the abstraction of two extreme prototypes on the continuous criterion dimension of the task. The prototypes are stored in memory as valid reference points for future similarity-based judgments. This prototype model was compared with the cue abstraction model, which assumes that people abstract cue weights in learning and add the cue information from exemplars to infer their criterion values varying on the continuous dimension. This study hypothesises that the training mode and the number of exemplars in training interact and affect subsequent model performance at test. The results from an experiment confirmed this hypothesis and showed that observational training supports an efficient prototype abstraction and feedback training supports an efficient cue abstraction.     

Dual processes,evolution and rationality

In three experiments we tested hypotheses derived from the goal specificity literature using a real-world physics task. In the balance-scale paradigm participants predict the state of the apparatus based on a configuration of weights at various distances from the fulcrum. Non-specific goals (NSG) have been shown to encourage hypothesis testing, which facilitates rule discovery, whereas specific goals (SG) do not. We showed that this goal specificity effect depends on task difficulty. The NSG strategy led to rule induction among some participants. Among non-discoverers, SG participants were faster and more accurate on difficult problems than NSG participants. The use of misleading exemplars (scale configurations that obscured the rule governing outcomes) led to fixation on inappropriate hypotheses for NSG but not SG participants. When more diagnostic learning exemplars were used, NSG non-discoverers still performed worse than SG participants on difficult problems. SG participants also outperformed NSG participants on a post-test of difficult problems. These findings qualify the generality of goal specificity effects.     

Cognitive skills in mathematical problem solving in Grade 3

Background. Research on the relationship between cognitive skills and mathematical problem solving is usually conducted on adults or on participants with acquired deficits associated with brain injury (e.g. Cipolotti, 1995 ; Cohen, Dehaene, & Verstichel, 1994 ; McCloskey, 1992 ). Aims. In these studies we wanted to make a contribution to the field of children's mathematical problem solving. The first aim of this study was to investigate whether mathematical problem solving in children is merely determined by semantic elaboration, as hypothesized in some of the models of adult processing (semantic hypothesis). In addition, we aimed to investigate whether there is a continuum from very good to very poor mathematical problem solving among children with mathematical learning disabilities showing immature cognitive skills (maturational lag hypothesis). Sample. The participants were 376 third graders and 107 second graders. Method. The internal structure of the data was analysed with a principal components analysis. In addition, two MANOVA were conducted to compare children with learning disabilities or problems with age‐matched and performance‐matched subjects. Results. Two components, a semantic and a non‐semantic one, were needed to account for an adequate fit of the dataset. In addition, children with mathematical learning disabilities had less‐developed cognitive skills compared with peers without learning disabilities, but they did not differ from younger children on seven of the nine cognitive skills. Conclusions. This study highlighted that children's mathematical problem solving is not determined by one general component. The picture is more complex, since two mathematics components were found. In addition, although our findings point in the direction of the maturational lag hypothesis it may be important to assess the different cognitive skills and especially assess the number system knowledge, since it seems below average in children with mathematical learning disabilities, compared with the knowledge of younger children with comparable skills in mathematics.     

The time course of implicit and explicit concept learning

The present experiment investigated the development of implicit and explicit knowledge during concept learning. According to Cleeremans and Jiménez (2002), the content of a representation can be conscious only when the representation is of a sufficiently good quality; on this theory, increasing explicit and decreasing implicit knowledge might be expected with training. The view that implicit knowledge arises from compilation of explicit knowledge makes the opposite prediction. The present research tested these possibilities using subjective measures based on confidence ratings. One group of participants was presented with blocks of category exemplars that activated prior knowledge, whereas another group was presented with blocks of categories that did not elicit any useful prior knowledge. The results showed that, irrespective of the knowledge group participants were allocated to, explicit knowledge increased over the course of learning, whereas implicit knowledge either stayed the same or decreased, consistent with Cleeremans and Jiménez's prediction.     

Arithmetic word problem solving: evidence for a magnitude-based mental representation

Previous findings have suggested that number processing involves a mental representation of numerical magnitude. Other research has shown that sensory experiences are part and parcel of the mental representation (or “simulation”) that individuals construct during reading. We aimed at exploring whether arithmetic word-problem solving entails the construction of a mental simulation based on a representation of numerical magnitude. Participants were required to solve word problems and to perform an intermediate figure discrimination task that matched or mismatched, in terms of magnitude comparison, the mental representations that individuals constructed during problem solving. Our results showed that participants were faster in the discrimination task and performed better in the solving task when the figures matched the mental representations. These findings provide evidence that an analog magnitude-based mental representation is routinely activated during word-problem solving, and they add to a growing body of literature that emphasizes the experiential view of language comprehension.     

Cue abstraction and exemplar memory in categorization

In this article, the authors compare 3 generic models of the cognitive processes in a categorization task. The cue abstraction model implies abstraction in training of explicit cue-criterion relations that are mentally integrated to form a judgment, the lexicographic heuristic uses only the most valid cue, and the exemplar-based model relies on retrieval of exemplars. The results from 2 experiments showed that, in lieu of the lexicographic heuristic, most participants spontaneously integrate cues. In contrast to single-system views, exemplar memory appeared to dominate when the feedback was poor, but when the feedback was rich enough to allow the participants to discern the task structure, it was exploited for abstraction of explicit cue-criterion relations.     

Self-regulated learning of a natural category: Do people interleave or block exemplars during study?

Despite decades of research focused on the representation of concepts, little is known about the influence of self-regulatory processes when learning natural categories. Such work is vital, as many contexts require self-regulation when we form complex concepts. Previous research has demonstrated that interleaving, as compared to blocking, can improve classification. Thus, as an initial step to explore self-regulated learning of natural concepts, we evaluated whether people chose to block or interleave their practice. According to the search-for-differences hypothesis, people attempt to identify features of birds that distinguish one category (i.e., bird family) from another, and hence should interleave their study. According to the search-for-similarities hypothesis, people attempt to identify features that indicate inclusion into a single category, and hence are expected to block their study. To evaluate these hypotheses, we had participants learn exemplar birds (e.g., Song Sparrow) with their respective bird families (e.g., Sparrow) by selecting the order in which to study bird families. Across four experiments, different formats for selecting exemplars for study were used, so as to provide converging evidence for how participants regulated their learning. Participants overwhelmingly preferred to block their study, even though interleaving is normatively better for learning.     

Episodic and prototype models of category learning

The question of what processes are involved in the acquisition and representation of categories remains unresolved despite several decades of research. Studies using the well-known prototype distortion task (Posner and Keele in J Exp Psychol 77:353–363, 1968) delineate three candidate models. According to exemplar-based models, we memorize each instance of a category and when asked to decide whether novel items are category members or not, the decision is explicitly based on a similarity comparison with each stored instance. By contrast, prototype models assume that categorization is based on the similarity of the target item to an implicit abstraction of the central tendency or average of previously encountered instances. A third model suggests that the categorization of prototype distortions does not depend on pre-exposure to study exemplars at all and instead reflects properties of the stimuli that are easily learned during the test. The four experiments reported here found evidence that categorization in this task is predicated on the first and third of these models, namely transfer at test and the exemplar-based model. But we found no evidence for the second candidate model that assumed that categorization is based on implicit prototype abstraction.     

The Power of a “Maverick” in Collaborative Problem Solving: An Experimental Investigation of Individual Perspective‐Taking Within a Group

Integrating different perspectives is a sophisticated strategy for developing constructive interactions in collaborative problem solving. However, cognitive aspects such as individuals’ knowledge and bias often obscure group consensus and produce conflict. This study investigated collaborative problem solving, focusing on a group member interacting with another member having a different perspective (a “maverick”). It was predicted that mavericks might mitigate disadvantages and facilitate perspective taking during problem solving. Thus, 344 university students participated in two laboratory‐based experiments by engaging in a simple rule‐discovery task that raised conflicts among perspectives. They interacted with virtual partners whose conversations were controlled by multiple conversational agents. Results show that when participants interacted with a maverick during the task, they were able to take others’ perspectives and integrate different perspectives to solve the problem. Moreover, when participants interacted in groups with a positive mood, groups with a maverick outperformed groups having several perspectives.