代数应用题项目生成中的认知过程与任务特征分析

认知项目设计起源于理论驱动的测量模式,要求在详尽阐述测量建构理论的基础上,鉴别测验项目类型的任务特征,建立和标定与测量建构认知变量之间的对应关系,从而通过系统变换任务特征生成项目.本文以Greeno (1994)提出的代理者-情境交互作用(agent-situation interaction)的问题解决观为基础,对代数应用题解决的认知过程和任务蕴含性(affordance)进行了系统分析.具体而言,代数应用题被视为以文本形式呈现的,嵌套在现实情境中的某种量化结构.在代数应用题解决的转译、整合、计划、执行的认知过程中需要形成三种不同的表征,即文本基、情境模型和问题模型.其形成及解决过程受到个体问题图式知识的深刻影响.对应于不同表征方式,对代数应用题语法/语义结构、量化元素及结构、问题情境特征进行了系统分析和综合.该分析表明,为了保证代数应用题项目生成的科学性,当前研究的关键在于建立任务特征与问题解决认知变量之间的经验关系,并在问题领域整体水平上检验这种关系的可推广性.     

Priming,analogy, and awareness in complex reasoning

The mechanisms by which a concept used in solving one complex task can influence performance on another complex task were investigated. We tested the hypothesis that even when subjects do not spontaneously make an analogy between two domains, knowledge of one domain can still spontaneously influence reasoning about the other domain via the mechanism of priming. Four groups of subjects (two experimental and two control) were given a simulated biochemistry problem on Day 1 and a simulated molecular genetics problem on Day 2. For the two experimental groups, the solution to the biochemistry problem involved inhibition. For the two control groups, the solution did not involve inhibition. On Day 2, all subjects received the same version of the molecular genetics problem in which the solution involved the concept of inhibition. Subjects in the experimental conditions were more likely to attain the correct answer, to propose inhibition, and to propose inhibition early in the problemsolving session than were subjects in the control conditions. However, subjects in the experimental conditions made no reference to the biochemistry problem either in their verbal protocols or in a posttask questionnaire. The results are interpreted as demonstrating that an implicit process—priming— can make old knowledge available for current problem solving.     

Electrifying diagrams for learning: principles for complex representational systems

Six characteristics of effective representational systems for conceptual learning in complex domains have been identified. Such representations should: (1) integrate levels of abstraction; (2) combine globally homogeneous with locally heterogeneous representation of concepts; (3) integrate alternative perspectives of the domain; (4) support malleable manipulation of expressions; (5) possess compact procedures; and (6) have uniform procedures. The characteristics were discovered by analysing and evaluating a novel diagrammatic representation that has been invented to support students' comprehension of electricity—AVOW diagrams (Amps, Volts, Ohms, Watts). A task analysis is presented that demonstrates that problem solving using a conventional algebraic approach demands more effort than AVOW diagrams. In an experiment comparing two groups of learners using the alternative approaches, the group using AVOW diagrams learned more than the group using equations and were better able to solve complex transfer problems and questions involving multiple constraints. Analysis of verbal protocols and work scratchings showed that the AVOW diagram group, in contrast to the equations group, acquired a coherently organised network of concepts, learnt effective problem solving procedures, and experienced more positive learning events. The six principles of effective representations were proposed on the basis of these findings. AVOW diagrams are Law Encoding Diagrams, a general class of representations that have been shown to support learning in other scientific domains.     

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.     

Prospective Adaptation in the Use of External Representations

An important element of adaptive expertise involves stepping away from a routine to retool one's knowledge or environment. The current study investigated two forms of this adaptive pattern: fault-driven adaptations, which are reactions to a difficulty, and prospective adaptations, which are proactive reformulations. Graduate and undergraduate students with no medical training engaged in a medical diagnosis task that involved complex information management. The graduate students, who were relative experts in information management and data analysis, uniformly made prospective adaptations by taking the time to create external representations of the available information before they diagnosed a single patient. In contrast, the undergraduate students only made representations reactively, when experimental manipulations made their default behaviors impractical. Graduate students tolerated the time lost creating representations in favor of future benefits—well-structured representations led to more optimal diagnostic choices. Overall, the results indicate that long-term educational experiences are correlated with prospective adaptation, even in a novel task domain that is not explicitly a part of those educational experiences. This research provides new metrics for evaluating educational interventions designed to move students along a trajectory toward adaptive expertise.     

The verbalization of multiple strategies in a variant of the traveling salesperson problem

What kinds of strategies do humans employ when confronted with a complex spatial task, and how do they verbalize these strategies? Previous research concerned with the well-known traveling salesperson problem (TSP) typically aimed at the identification of a generally applicable heuristics that adequately represents human behavior in relation to the abstract task of combining points. This paper adopts a novel perspective in two respects. On the one hand, it addresses the strategies people employ when confronted with a more complex task, involving distractors and feature information rather than identical points. On the other hand, retrospective linguistic representations of the strategies used are analyzed in relation to the behavioral data, using discourse analytic methods. Results show that both the behavioral results and the verbalizations point to a range of strategies related to those proposed for solving abstract TSPs. However, in contrast to earlier accounts in the literature, the participants employ a repertory of multi-faceted strategies and planning processes, simplifying and structuring the problem space across subtasks and processes in flexible ways. These findings provide further insight into the nature of human strategies in spatial problem solving tasks and their retrospective verbalization, highlighting how procedures generally known in the literature may be adapted to more complex tasks, and how they may be verbalized spontaneously.     

Expertise as mental set: The effects of domain knowledge in creative problem solving

Experts generally solve problems in their fields more effectively than novices because their wellstructured, easily activated knowledge allows for efficient search of a solution space. But what happens when a problem requires a broad search for a solution? One concern is that subjects with a large amount of domain knowledge may actually be at a disadvantage, because their knowledge may confine them to an area of the search space in which the solution does not reside. In other words, domain knowledge may act as a mental set, promoting fixation in creative problem-solving attempts. A series of three experiments in which an adapted version of Mednick’s (1962) remote associates task was used demonstrates conditions under which domain knowledge may inhibit creative problem solving.     

Dissociable stages of problem solving (II): first evidence for process-contingent temporal order of activation in dorsolateral prefrontal cortex

In a companion study, eye-movement analyses in the Tower of London task (TOL) revealed independent indicators of functionally separable cognitive processes during problem solving, with processes of building up an internal representation of the problem preceding actual planning processes. These results imply that processes of internalization and planning should also be distinguishable in time and space with respect to concomitant brain activation patterns. To investigate this possibility, here we conducted analyses of fMRI data for left and right dorsolateral prefrontal cortex (dlPFC) during problem solving in the TOL task by accounting for the trial-by-trial variability of onsets and durations of the different cognitive processing stages. Comparisons between stimulus-locked and response-locked modeling approaches affirmed that activation in left dlPFC was elicited particularly during early processes of internalization, comprising the extraction of goal information and the generation of an internal problem representation, whereas activation in right dlPFC was predominantly attributable to later processes of mental transformations on this representation, that is planning proper. Thus, present data corroborate the proposal that often observed bilateral dlPFC activation patterns during complex cognitive tasks such as problem solving may reflect functionally and, to some extent, even temporally separable processes with opposing lateralizations.     

Analogy as a strategy for supporting complex problem solving under uncertainty

Complex problem solving in naturalistic environments is fraught with uncertainty, which has significant impacts on problem-solving behavior. Thus, theories of human problem solving should include accounts of the cognitive strategies people bring to bear to deal with uncertainty during problem solving. In this article, we present evidence that analogy is one such strategy. Using statistical analyses of the temporal dynamics between analogy and expressed uncertainty in the naturalistic problem-solving conversations among scientists on the Mars Rover Mission, we show that spikes in expressed uncertainty reliably predict analogy use (Study 1) and that expressed uncertainty reduces to baseline levels following analogy use (Study 2). In addition, in Study 3, we show with qualitative analyses that this relationship between uncertainty and analogy is not due to miscommunication-related uncertainty but, rather, is primarily concentrated on substantive problem-solving issues. Finally, we discuss a hypothesis about how analogy might serve as an uncertainty reduction strategy in naturalistic complex problem solving.     

Goal specificity and knowledge acquisition in statistics problem solving: Evidence for attentional focus

Solving training problems with nonspecific goals (NG; i.e., solving for all possible unknown values) often results in better transfer than solving training problems with standard goals (SG; i.e., solving for one particular unknown value). In this study, we evaluated an attentional focus explanation of the goal specificity effect. According to the attentional focus view, solving NG problems causes attention to be directed to local relations among successive problem states, whereas solving SG problems causes attention to be directed to relations between the various problem states and the goal state. Attention to the former is thought to enhance structural knowledge about the problem domain and thus promote transfer. Results supported this view because structurally different transfer problems were solved faster following NG training than following SG training. Moreover, structural knowledge representations revealed more links depicting local relations following NG training and more links to the training goal following SG training. As predicted, these effects were obtained only by domain novices.     

小学生视觉-空间表征类型和数学问题解决的研究

本研究考察并比较了四至六年级儿童的视觉－空间表征策略、数学问题解决和空间视觉化能力。结果表明：五、六年级儿童的解题正确率、使用图式表征策略的程度显著高于四年级儿童;使用图像表征策略的程度各年级无显著差异。将数学问题分成三个难度等级,发现年级差异主要表现在难度等级1的题目上。另外,六年级儿童的空间视觉化能力显著高于四年级儿童。     

Learning representations in a gated prefrontal cortex model of dynamic task switching

The prefrontal cortex is widely believed to play an important role in facilitating people's ability to switch performance between different tasks. We present a biologically‐based computational model of prefrontal cortex (PFC) that explains its role in task switching in terms of the greater flexibility conferred by activation‐based working memory representations in PFC, as compared with more slowly adapting weight‐based memory mechanisms. Specifically we show that PFC representations can be rapidly updated when a task switches via a dynamic gating mechanism based on a temporal‐differences reward‐prediction learning mechanism. Unlike prior models of this type, the present model develops all of its internal representations via learning mechanisms as shaped by the demands of continuous periodic task switching. This advance opens up a new domain of research into the interactions between working memory task demands and the representations that develop to meet them. Results on a version of the Wisconsin card sorting task are presented for the full model and a number of comparison networks that test the importance of various model features. Furthermore, we show that a lesioned model produces perseverative errors like those seen in frontal patients.     

Cognitively active externalization for situated reflection

This paper offers an explanation of how collaboration leads to abstract and flexible problem solving. We asked the individual and paired subjects to indicate 3/4 of 2/3 of the area of a square sheet of paper and found that (1) they primarily folded or partitioned the paper rather than algorithmically calculating the answer, (2) they strongly tendened to backtrack and confirm their proto‐plans on externalized traces such as creases on the paper, and (3) only the paired subjects shifted to the mathematical strategy in their second trials. Based on these results, we propose that two factors, individuals' activeness in choosing and confirming the initial strategies and the frequent role exchange between task‐doing and monitoring in collaborative situations, interact in collaboration to generate various solutions differing in the degree of abstraction, which are then reflected upon by the participants to lead them to abstraction.     

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.     

Working Wonders? Investigating insight with magic tricks

We propose a new approach to differentiate between insight and noninsight problem solving, by introducing magic tricks as problem solving domain. We argue that magic tricks are ideally suited to investigate representational change, the key mechanism that yields sudden insight into the solution of a problem, because in order to gain insight into the magicians’ secret method, observers must overcome implicit constraints and thus change their problem representation. In Experiment 1, 50 participants were exposed to 34 different magic tricks, asking them to find out how the trick was accomplished. Upon solving a trick, participants indicated if they had reached the solution either with or without insight. Insight was reported in 41.1% of solutions. The new task domain revealed differences in solution accuracy, time course and solution confidence with insight solutions being more likely to be true, reached earlier, and obtaining higher confidence ratings. In Experiment 2, we explored which role self-imposed constraints actually play in magic tricks. 62 participants were presented with 12 magic tricks. One group received verbal cues, providing solution relevant information without giving the solution away. The control group received no informative cue. Experiment 2 showed that participants’ constraints were suggestible to verbal cues, resulting in higher solution rates. Thus, magic tricks provide more detailed information about the differences between insightful and noninsightful problem solving, and the underlying mechanisms that are necessary to have an insight.     

A rational analysis of the acquisition of multisensory representations

How do people learn multisensory, or amodal, representations, and what consequences do these representations have for perceptual performance? We address this question by performing a rational analysis of the problem of learning multisensory representations. This analysis makes use of a Bayesian nonparametric model that acquires latent multisensory features that optimally explain the unisensory features arising in individual sensory modalities. The model qualitatively accounts for several important aspects of multisensory perception: (a) it integrates information from multiple sensory sources in such a way that it leads to superior performances in, for example, categorization tasks; (b) its performances suggest that multisensory training leads to better learning than unisensory training, even when testing is conducted in unisensory conditions; (c) its multisensory representations are modality invariant; and (d) it predicts ‘‘missing” sensory representations in modalities when the input to those modalities is absent. Our rational analysis indicates that all of these aspects emerge as part of the optimal solution to the problem of learning to represent complex multisensory environments.     

代数应用题项目生成的结构分析方法

自动化项目生成是近年来兴起的测量领域, 是一种以项目认知加工理论为基础的原则性项目设计(principled item design)模式。其中, 如何在项目认知模型基础上, 通过任务结构分析的方式系统全面的鉴别和提取任务特征是一个关键环节。基于已有文献中代数应用题的命题分析法、网络语言分析法、关系-函数分析法、任务分析地图等四种结构分析方法, 研究探索了能够服务于自动化项目生成的代数应用题任务结构分析方法。该分析表明, 前三种方法分别对应于个体解题过程需要形成的三种中介表征, 即问题陈述背后的命题表征、事件时空关系的情境模型、以及变量间数量关系的问题模型, 第四种方法从过程角度分析了问题解决的认知需求。然而, 要实现项目生成的特征提取需求, 尚需对现有四种方法所揭示问题特征的心理现实性、特征提取的系统性和完备性、任务领域的适用范围、以及不同方法的整合等问题开展进一步研究。     

Seeing the Problem: An Explanation From Pólya

In this article, we focus on the instructional explanation of guessing as a heuristic for solving the Five Planes Problem (FPP), given in a lesson by George Pólya. We use the analysis of the lesson to test the applicability of what is known about instructional explanations in elementary mathematics to higher mathematics. The most salient characteristic of Pólya's lesson is the profusion of models and representations used to develop a sense of the problem and to support the instructional explanation. Pólya used analogical models to transform the complex FFT to a simpler one, and he used representations to extend the perspective on the problem. Introducing such models and representations requires keeping track of the links between them and the original problem. Pólya excelled in this endeavor, and his passage to each new representation or model was generally justified in terms of the goals of the explanation. Another very important feature of an instructional explanation is problem identification, a fragile goal state that needs to be constantly maintained when the problem being explained is complex. For this lesson, we examine how Pólya established and maintained that goal. Finally, we offer some insight on instructional explanations in a situation in which the teacher needs to fulfill two goals at the same time. Here, the first goal was to teach students both how to solve FPP and how to use guessing as a problem-solving heuristic or strategy; the second goal was to show us how to teach guessing. Keeping these goals in mind, we offer some suggestions on how to teach metaskills through mathematical problems in accordance with the current understanding of constructivist approaches to learning.     

Risky Communication: Pitfalls in Counseling About Risk,and How to Avoid Them

A genetic counselor is often faced with the difficult task of conveying a set of complex and highly abstract factors associated with the client's risk of developing a familial disorder. The client is faced with the even more difficult task of making significant health-related decisions about an event which may or may not eventuate. Although there is a large corpus of research on this topic, much of the knowledge on risk communication is difficult to apply in a practical context. In this paper we draw together some insights on risk communication and decision-making under conditions of uncertainty, and apply them directly to the problem of communicating familial cancer risk. In particular, we focus on the distinction between individual risk and observed frequencies of adverse events, various framing effects, and contextualizing risk communication. We draw attention to some of the potential pitfalls in counseling about risk and offer avenues for circumventing them.