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.     

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.     

以"问题解决"为突破口,优化生物学教学

将“问题解决”（problem solving)这一新的教育理念，引入到生物学课堂中，对其可行性进行初步探讨，优化生物学教学，培养学生分析问题，解决问题的能力。     

How handwriting behaviors during problem solving are related to problem-solving success in an engineering course

If we carefully observe the spatial and temporal organization of students' pen strokes as they solve an engineering problem, can we predict their ability to achieve the correct answer? To address this question, 122 college students were asked to solve exam problems in an engineering course using a smartpen that recorded their writing as digitized timestamped pen strokes. The pen stroke data was used to compute a collection of 10 metrics characterizing various elements of problem-solving fluency including the tendency to progress down the page without revisions, the amount of time with no activity, and the frequency of constructing and using equations. The primary finding is that, on average across 13 different exam problems, these elements of problem-solving process explained 40% of the variance in scores of the correctness of the problem solution. In short, success on generating correct solutions was related to the fluency of the student's problem-solving process (i.e., working sequentially from the top to the bottom of the page, working without detours or long pauses, and working by constructing equations). This work is consistent with the idea that expertise in solving common engineering problems involves being able to treat them like routine rather than non-routine problems.     

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。     

化学教学中问题解决法初探

问题解决是思维的最一般形式,是人类适应环境、解决生存与发展中各种问题的基本方式。实际的教学中则是通过创设问题情境,在教学内容与学生的求知心理之间建立一种不平衡状态,把学生引入到与问题有关的情境之中。利用此模式的教学是改变以被动接受为主的教学模式,实行探究学习与合作学习的重要途径。     

Linking complex problem solving to opportunity identification competence within the context of entrepreneurship

Today’s working life is increasingly characterized by entrepreneurial challenges. Entrepreneurial challenges start at an individual level with the identification of entrepreneurial opportunities, which is acknowledged as one of the key competencies for lifelong learning. Since the identification of entrepreneurial opportunities relies heavily on the opportunity identification competence (OIC) of individuals, understanding the meaning of OIC is relevant. Until now, OIC has been explored in the young entrepreneurship research field. However, entrepreneurship researchers until now have not fully explored OIC. According to several authors, the research on complex problem solving (CPS) in the cognitive research field might contribute to understanding OIC. In this paper, we review the link between OIC and CPS by comparing the cognitive and entrepreneurship research fields. We argue that those who excel in identifying opportunities share core characteristics with high-level complex problem-solvers. We propose to conduct empirical research in the future to investigate the relation between OIC and CPS within a work context in order to gain more insight into OIC. We believe that the cognitive research field contributes to the entrepreneurship research field and provides a deeper understanding of the initial steps of the entrepreneurial process.     

Non‐mathematical problem solving in organic chemistry

Differences in problem‐solving ability among organic chemistry graduate students and faculty were studied within the domain of problems that involved the determination of the structure of a molecule from the molecular formula of the compound and a combination of IR and ¹H NMR spectra. The participants' performance on these tasks was compared across variables that included amount of research experience, year of graduate study, and level of problem‐solving confidence. Thirteen of the 15 participants could be classified as either “more successful” or “less successful.” The participants in this study who were “more successful” adopted consistent approaches to solving the problems; were more likely to draw molecular fragments obtained during intermediate stages in the problem‐solving process; were better at mining the spectral data; and were more likely to check their final answer against the spectra upon which the answer was based. Experience from research, teaching, and course work were found to be important factors influencing the level of participants' success. © 2009 Wiley Periodicals, Inc. J Res Sci Teach 47:643–660, 2010     

Junior high school physics: Using a qualitative strategy for successful problem solving

Students at the junior high school (JHS) level often cannot use their knowledge of physics for explaining and predicting phenomena. We claim that this difficulty stems from the fact that explanations are multi‐step reasoning tasks, and students often lack the qualitative problem‐solving strategies needed to guide them. This article describes a new instructional approach for teaching mechanics at the JHS level that explicitly teaches such a strategy. The strategy involves easy to use visual representations and leads from characterizing the system in terms of interactions to the design of free‐body force diagrams. These diagrams are used for explaining and predicting phenomena based on Newton's laws. The findings show that 9th grade students who studied by the approach advanced significantly from pretests to post‐tests on items of the Force Concept Inventory—FCI and on other items examining specific basic and complex understanding performances. These items focused on the major learning goals of the program. In the post‐tests the JHS students performed on the FCI items better than advanced high‐school and college students. In addition, interviews conducted before, during, and after instruction indicated that the students had an improved ability to explain and predict phenomena using physics ideas and that they showed retention after 6 months. © 2010 Wiley Periodicals, Inc. J Res Sci Teach 47: 1094–1115, 2010     

The contribution of external representations in pre-school mathematical problem solving

Could problem solving be the object of teaching in early education? Could appropriate teaching interventions develop to scaffold children's efforts to solve problems? These were the central questions of this article. The sample consisted of 18 children attending public pre-school in Cyprus. The problem they were asked to solve was to find all solutions of the pentomino. The children's problem solving was supported by graphically representing their solutions on squared paper. The findings show that children responded positively to the problem and were successful in finding all solutions for the specific problem. The graphical representation of the solutions and the forms of teacher–children and children–children interactions played an important role in the positive outcome of the activity.     

The effects of knowledge,strategies, and the interaction between the two in verbal analogy problem solving

An experiment was conducted to assess how accessibility to relevant knowledge would affect performance on verbal analogy problems as well as the efficacy, perception, and continuation of use of a componential strategy for solving these problems. Participants were either taught to use a componential strategy for solving verbal analogy problems or just given practice opportunities while the accessibility of relevant knowledge was manipulated and the performance and perception of the strategy was assessed. We also investigated whether variations in accessibility to relevant knowledge affected the continuance of strategy use. Results showed that knowledge accessibility affected both performance and the way in which the strategy was perceived. Participants rated the strategy as more useful, easier to use, and less effortful when their initial strategy experience involved solving analogy problems based on familiar vocabulary and relationships. Also, participants who had experience with easily accessible vocabulary and relationships were significantly more likely to choose to continue to use of the strategy on a subsequent task. It is also argued that accessibility to knowledge and strategy use play independent, as opposed to interactive, roles in determining performance.     

Individual differences in memory updating in relation to arithmetic problem solving

The study investigates the relationship between memory updating and arithmetic word problem solving. Two groups of 35 fourth graders with high and low memory-updating abilities were selected from a sample of 89 children on the basis of an updating task used by Palladino et al. [Memory & Cognition 29 (2002) 344]. The two groups were required to solve a set of arithmetic word problems and to recall relevant information from another set of problems. Several span tasks, a computation test, and the PMA verbal subtest were also administered. The group with a high memory-updating ability performed better in problem solving, recalling text problems, and in the computation test. The two groups did not differ in the PMA verbal subtest or in the digit and word spans. Results were interpreted as supporting the importance of updating ability in problem solving and of the substantial independence between memory updating and problem solving on one hand and verbal intelligence on the other.     

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.     

Assessment of expert-novice chemistry problem solving using HyperCard: Early findings

Results of a HyperCard method for assessing the performance of expert and novice high school chemistry students solving stoichiometric chemistry problems (balancing chemical equations) have been reported. The assessment involved the use of a HyperCard instrument (Hyperequation) developed on a Macintosh platform to administer a set of five chemistry problems, register student responses, and collect data related to student performance. The chemistry problems have been previously tested and validated in traditional (pen-paper) assessment settings by other researchers. MANOVA results indicate a significant difference between the performance of expert and novice students solving the five stoichiometric chemistry problems using the Hyperequation. The study shows promise in that a HyperCard assessment method could differentiate between the performance of experts and novices in problem solving. The implication is that HyperCard might be a suitable technology for developing performance assessment methods not only in chemistry but also in other science disciplines.     

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.     

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.     

Manipulatives and number sentences in computer aided arithmetic word problem solving

In this study we investigated the relative contribution of two main components often used in the instruction of arithmetic word problem solving to first-grade children and children with learning problems, external representation with manipulatives and formal mathematical representation with number sentences. Four computer aided treatments were developed along these dimensions. Furthermore, an assessment control group was formed.It was shown that performance improved significantly after treatment in comparison to the performance of the assessment control group. Comparison of the different training conditions showed that a number sentence treatment was the most effective training component. However, this effect was only present with the relatively high competent children and not with the relatively low performing children.     

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。     

Fostering analogical transfer: The multiple components approach to algebra word problem solving in a chemistry context

Holyoak and Koh (1987) and Holyoak (1984) propose four critical tasks for analogical transfer to occur in problem solving. A study was conducted to test this hypothesis by comparing a multiple components (MC) approach against worked examples (WE) in helping students to solve algebra word problems in chemistry classes. The MC approach incorporated multiple components (symbolic equations, symbols, categorization, hint) in the source, or target, or both, to address the four analogical tasks. Different combinations of the components were tested in a series of four experiments. Symbolic equations (main component) fostered a mental construction of the problem in its solution mode. Categorization enabled an identification of the problem category. A hint in the target directed the learners to the source problem. The interaction between these components facilitated the mapping of the symbolic equations in the source onto the target, resulting in the superiority of the MC approach in fostering analogical transfer. Neither the main component alone nor the main component plus one sub-component was sufficient for analogical transfer. Hence for analogical transfer to occur, at least the main component (symbolic equations) and two sub-components (categorization and hint) are required. However, symbols may not have additional effects for transfer to occur.     

The consistency effect in word problem solving is effectively reduced through verbal instruction

In mathematical word problem solving, a relatively well-established finding is that more errors are made on word problems in which the relational keyword is inconsistent instead of consistent with the required arithmetic operation. This study aimed at reducing this consistency effect. Children solved a set of compare word problems before and after receiving a verbal instruction focusing on the consistency effect (or a control verbal instruction). Additionally, we explored potential transfer of the verbal instruction to word problems containing other relational keywords (e.g., larger/smaller than) than those in the verbal instruction (e.g., more/less than). Results showed a significant pretest-to posttest reduction of the consistency effect (but also an unexpected decrement on marked consistent problems) after the experimental verbal instruction but not after the control verbal instruction. No significant effects were found regarding transfer. It is concluded that our verbal instruction was useful for reducing the consistency effect, but future research should address how this benefit can be maintained without hampering performance on marked consistent problems.