Teaching algorithmic problem solving or conceptual understanding: Role of developmental level,mental capacity,and cognitive style

It has been shown previously that many students solve chemistry problems using only algorithmic strategies and do not understand the chemical concepts on which the problems are based. It is plausible to suggest that if the information is presented in differing formats, the cognitive demand of a problem changes. The main objective of this study is to investigate the degree to which cognitive variables, such as developmental level, mental capacity, and disembedding ability explain student performance on problems which: (1) could be addressed by algorithms or (2) require conceptual understanding. All conceptual problems used in this study were based on a figurative format. The results obtained show that in all four problems requiring algorithmic strategies, developmental level of the students is the best predictor of success. This could be attributed to the fact that these are basically computational problems, requiring mathematical transformations. Although all three problems requiring conceptual understanding had an important aspect in common (the figurative format), in all three the best predictor of success is a different cognitive variable. It was concluded that: (1) the ability to solve computational problems (based on algorithms) is not the major factor in predicting success in solving problems that require conceptual understanding; (2) solving problems based on algorithmic strategies requires formal operational reasoning to a certain degree; and (3) student difficulty in solving problems that require conceptual understanding could be attributed to different cognitive variables.     

Imitative problem solving: Why transfer of learning often fails to occur

The paper presents empirical evidence for imitativeproblem solving and the Interpretation Theorydescribed in Robertson and Kahney (1996). According tothe theory beginners use imitation as their primary problem solving method when learning about an unfamiliar domain. Imitative problem solving can explain much of the evidence that analogical transfereven within a domain is often hard to find. The paperpresents an analysis of algebra word problems topredict in detail exactly where solvers will havedifficulty in using a worked out example to solveeither a close or distant variant of the problem type.In a 2 × between-groups design, secondary schoolstudents were given an explanation of an algebra wordproblem taken from Reed et al. (1985)or an explanation of the problem that also includedinformation about how the solution could be adapted tosolve a distant variant. They were then given eithera close or distant variant to solve. Results were inline with the predictions derived from theInterpretation Theory analysis.     

有关数学问题的解题思想与方法

我们从中学就开始接触各类数学问题,而要解决这些数学问题,最重要的就是找出问题的精髓也就是所运用的思想与方法,并且这些思想与方法在实际应用中也非常广泛,因此,在这里我们主要介绍几种重要的解决数学问题的思想与方法。     

引入 AD 理论的产品设计课程建设研究

探索基于当地合作企业的产品开发需求和技术资源,引入更符合人们认知和思维方式、更具工程控制性的公理设计（AD）到产品设计课程之中来,通过工作坊模式来开展课堂教学,解决工业设计专业课程中理论与实践结合问题,形成产、学、研一体,课内教学与课外实践相结合的教学模式。     

Creativity and inspiration for problem solving in engineering education

Problem solving is a critical skill for engineering students and essential to development of creativity and innovativeness. Essential to such learning is an ease of communication and allowing students to address the issues at hand via the terminology, attitudes, humor and empathy, which is inherent to their frame of mind as novices, without the attempt to have to be the expert. Deep learning of scientific fact can be facilitated by using non-conventional tools for teaching, learning and presentation such as drama, video, posters, model making and other similar means. It may be time to break free of the PowerPoint tradition to generate successful approaches for establishing student engagement and maintaining such engagement.     

Computer technology and complex problem solving: Issues in the study of complex cognitive activity

Goals and plans organize much of complex problem solving behavior and are often inferable from action sequences. This paper addresses the strengths and limitations of inferring goals and plans from information that can be derived from computer traces of software used to solve mathematics problems. We examined mathematics problem solving activity about distance, rate, time relationships in a computer software environment designed to support understanding of functional relationships among these variables (e.g., distance =rate × time; time=distance/rate) using graphical representations of the results of simulations. Ten adolescent-aged students used the software to solve two distance, rate, time problems, and provided think-aloud protocols. To determine the inferability of understanding from the action traces, coders analyzed students' understanding from the computer traces alone (Trace-only raters) and compared these to analyses based on the traces plus the verbal protocols (Traceplus raters). Inferability of understanding from the action traces was related to level of student understanding how they used the graphing tool. When students had a good understanding of distance, rate, time relationships, it could be accurately inferred from the computer traces if they used the simulation tool in conjunction with the graphing tool. When students had a weak understanding, the verbal protocols were necessary to make accurate inferences about what was and was not understood. The computer trace also failed to capture dynamic exploration of the visual environment so students who relied on the graphing tool were inaccurately characterized by the Trace-only coders. Discussion concerns types of scaffolds that would be helpful learning environment for complex problems, the kind of information that is needed to adequately track student understanding in this content domain, and instructional models for integrating learning environments like these into classrooms.Members of the Cognition and Technology Group at Vanderbilt who have contributed to this project are (in alphabetical order) Helen Bateman, John Bransford, Thaddeus Crews, Allison Moore, Mitchell Nathan, and Stephen Owens. The research was supported, in part, by grants from the National Science Foundation (NSF-MDR-9252990) but no official endorsement of the ideas expressed herein should be inferred.     

Chemistry problem solving instruction: a comparison of three computer-based formats for learning from hierarchical network problem representations

Within the cognitive load theory framework, we designed and compared three alternative instructional solution formats that can be derived from a common static hierarchical network representation depicting problem structure. The interactive-solution format permitted students to search in self-controlled manner for solution steps, static-solution format displayed all solutions steps, and no-solution format did not have solution steps. When we matched instructional time across the formats, in relation to the complex molarity problems rather than the dilution problems, differential transfer performance existed between the static-solution or no-solution formats and the interactive-solution format, but not between the static-solution format and no-solution format. The manner in which learners interact with the static-solution and no-solution formats depends on their level of expertise in the chemistry domain. With considerable learner expertise, provision of solution steps may be redundant incurring extraneous cognitive load. Absence of the solution steps may not have left sufficient cognitive capacity for germane cognitive load as some beginning learners lacked the prior knowledge to deduce the solution steps. Searching for solution steps presumably incurred extraneous cognitive load which interfered with learning and hence, in the interactive-solution format, it outweighed the benefit of engaging in self-regulated interaction with the content. Hence, cognitive load theory is a promising tool to predict the mental load associated with learning from the three alternative computer-based instructional formats.     

PISA2012问题解决评估框架分析及其对教学改革的启示*

“国际学生评估项目”（PISA）是经济合作与发展组织（OECD）为应答学生学业成绩的跨国比较研究需要而发动的,它着重评价青少年怎样在新的情境中运用知识和技能,即运用知识和技能迎接现实生活挑战的能力。“问题解决能力”的测评是2003年PISA评估中的附加内容,2012年再次把问题解决作为PISA评估项目之一。本文试图以OECD公布的PISA2012问题解决能力框架和方法、样题来反观我们的课堂,以期能为一线中小学教师的课堂教学提供一点思路和操作性建议。     

中美学生数学问题解决的差异比较

“问题解决”是国际数学教育研究的一个热点,本文从中美两国学生在问题解决的过程和结果的比较研究入手,比较和研究了中美两国学生在问题解决中提出问题的能力及解决问题的策略两方面的差异。     

The design and evaluation of hypertext structures for supporting design problem solving

Computer-based complex information systems are used increasingly more often, for a growing variety of purposes, in both educational and professional contexts. Since the effectiveness of information systems will largely depend on the particular purpose and the particular task context at hand, at least part of our research efforts should be directed at studying specific application areas. This paper reports a study on the use of hypertext information systems during architectural-design problem solving. Theoretical notions on design problem solving, such as distinguishing between a problem-structuring and a problem-solving phase, provide us with expectations about the changing informational needs during the design process. Specific information structures are proposed, incorporating design principles from learning research, to accommodate these informational needs. Results of an empirical study indeed showed interactions between design phase and information structure when separately inspecting the outcomes for problem structuring and problem solving. Educational implications include the use of a combination of hierarchical decomposition and cross-referencing for certain instructional goals, such as teaching complexity and abstraction.     

多Agent合作问题求解方法的研究综述

多Agent合作问题求解是多Agent系统理论与技术研究的重点.多Agent合作的目的是使合作求解的效用大于单个Agent独立完成的效用,因此多Agent之间如何协调一致,形成统一的行动步调或者规范也成为了多Agent合作问题求解的主要目的.介绍了目前研究多Agent合作问题求解的主要方法、多agent合作问题求解的步骤以及其应用领域,并给出了目前存在的问题与未来的发展方向.     

蜘蛛侠与蜘蛛精：隐喻的概念合成空间

认知语言学的隐喻研究框架包括概念隐喻理论和概念合成理论。前者指出思维的隐喻性,而后者则强调隐喻在不同域之间的合成方式。＂蜘蛛侠＂和＂蜘蛛精＂的输入空间相似,但输出结果却蕴含着不同的隐喻,价值观棱镜起到了语义核查的作用。     

Epistemic profiles and self-regulated learning: Examining relations in the context of mathematics problem solving

Relations were examined between epistemic profiles, regulation of cognition, and mathematics problem solving. Two hundred sixty-eight students were sampled from undergraduate mathematics and statistics courses. Students completed inventories reflecting their epistemic profiles and learning strategies, and were profiled as rational, empirical, or both. Based on their profiles, 24 students participated in two problem-solving sessions. Episodes were coded for planning, monitoring, control, use of empirical and rational argumentation, and justification for solutions. For both self-reported metacognitive self-regulation and regulation of cognition during problem solving, students profiled as rational had the highest self-reported mean and actual frequency of regulation of cognition compared to students profiled as predominantly empirical. Moreover, students profiled as predominantly rational correctly solved more problems than the other two groups. Finally, students’ approaches to problem solving were consistent with their epistemic profiles. Relations are discussed in the context of various theoretical frameworks.     

Designing and implementing a case-based learning environment for enhancing ill-structured problem solving: classroom management problems for prospective teachers

This design-based research study is aimed at two goals: (1) developing a feasible case-based instructional model that could enhance college students’ ill-structured problem solving abilities, while (2) implementing the model to improve teacher education students’ real-world problem solving abilities to deal with dilemmas faced by practicing teachers in elementary classrooms. To achieve these goals, an online case-based learning environment for classroom management problem solving (CBL-CMPS) was developed based on Jonassen’s (in: Reigeluth (ed.) Instructional-Design Theories and Models: A New Paradigm of Instructional Theory, 1999) constructivist learning environment model and the general process of ill-structured problem solving (1997). Two successive studies, in which the effectiveness of the CBL-CMPS was tested while the CBL-CMPS was revised, showed that the individual components of the CBL-CMPS promoted ill-structured problem solving abilities respectively, and that the CBL-CMPS as a whole learning environment was effective to a degree for the transfer of learning in ill-structured problem solving. The potential, challenge, and implications of the CBL-CMPS are discussed.

数学解题模式教学与数学能力的培养

通过对数学解题模式理论价值的探讨 ,揭示了数学解题模式教学与数学能力培养的关系 .     

Examining the unique contributions and developmental stability of individual forms of relational reasoning to mathematical problem solving

Relational reasoning, a higher-order cognitive ability that identifies meaningful patterns among information streams, has been suggested to underlie STEM development. This study attempted to explore the potentially unique contributions of four forms of relational reasoning (i.e., analogy, anomaly, antinomy, and antithesis) to mathematical problem solving. Two separate samples, fifth graders (n = 254) and ninth graders (n = 198), were assessed on their mathematical problem solving ability and the different forms of relational reasoning ability. Linear regression analysis was conducted, with participants’ age, working memory, and spatial skills as covariates. The results showed that analogical and antithetical reasoning abilities uniquely predicted mathematical problem solving. This pattern demonstrated developmental stability across a four-year time frame. The findings clarify the unique significance of individual forms of relational reasoning to mathematical problem solving and call for a shift of research direction to reasoning abilities when exploring dissimilarity-based relations (opposites in particular).     

The role of guidance in computer-based problem solving for the development of concepts of logic

The effect of two instructional variables,manipulation of objects and guidance, inlearning to use the logical connective,conditional, was investigated. Instructions for72 first- and second year social sciencestudents were varied in the computer-basedlearning environment Tarski's World, designedfor teaching first-order logic (Barwise &Etchemendy, 1992). Guidance, which wasoperationalised by giving the learners problemsthat guided them to all different types ofbasic problem situations that could be derivedfrom the conditional by stimulating them tomanipulate or to imagine to manipulate thegeometrical objects, significantly influencedthe scores from pre- to post test.Manipulation, which was operationalised bygiving the learners a visual representation inwhich (concrete) geometrical objects could bemanipulated, yielded significant results frompost- to retention test. The results supportthe authors' view that guidance in combinationwith the possibility to manipulate objects in adomain, support the acquisition of knowledgeand skills in that domain. Guidance leads tothe students seeing all situations that arerelevant for the development of new knowledge,which has a positive effect directly afterinstruction. The possibility to manipulateleads to the students playing around in such away that they (systematically) experience theresults of their actions by the feedback given,having an effect after two weeks ofnon-instruction.     

Tracing the development of representational flexibility and problem solving in fraction addition: a longitudinal study

The study models the development of students’ multiple-representation flexibility and the use of problem-solving strategies and representations in fraction addition. A test administered three times, with breaks of 3–4?months between successive measurements to 108 students at a transition within primary school (Grade 5–6), 132 students at a transition from primary to secondary education (Grade 6–7) and 148 students at a transition within secondary school (Grade 7–8). Multivariate analysis of variance for repeated measures and dynamic structural equation modelling were carried out in order to analyse the data. Findings suggested that students’ performance improve through measurements. Dynamic modelling provided evidence for the strong interrelation between representational flexibility and problem solving at the three measurements. The results indicated the students’ established pre-existent knowledge and the important role the initial state of the aforementioned cognitive parameters plays on their advancement. Didactical implications are discussed.     

Children's emotions in math problem solving situations: Contributions of self-concept,metacognitive experiences,and performance

This study examined 1) experiences of six discrete emotions ‒ joy, pride, contentment, worry, shame, hopelessness ‒ after solving a math problem of students aged 10–12 years, and 2) the contribution of self-concept, metacognitive experiences (feeling of difficulty and feeling of success) and performance on emotions experienced after the task, controlling for gender and emotions experienced before the task. Results indicated a decrease in joy and contentment after problem solving. Performance did not contribute to emotions apart from hopelessness. The influence of performance on hopelessness was mediatized by metacognitive experiences. Self-concept contributed to joy, pride and shame but its influence became non-significant when we controlled for metacognitive experiences. Feeling of success mediatized the effect of self-concept on joy, pride and shame. Metacognitive experiences were also found to be important predictors of all emotions except worry. The need for new paradigms to study emotions in education is discussed.