Teaching for transfer of core/key skills in higher education: Cognitive skills

This article is a result of a completed survey of the mainly cognitive science literature on the transferability of those skills which have been described variously as ‘core’, ‘key’, and ‘generic’. The literature reveals that those predominantly cognitive skills which have been studied thoroughly (mainly problem solving) are transferable under certain conditions. These conditions relate particularly to the methods and environment of the learning of these skills. Therefore, there are many implications for the teaching of key skills in higher education, which the article draws out, following a summary of the main findings of the research literature. Learning of principles and concepts facilitates transfer to dissimilar problems, as it creates more flexible mental representations, whereas rote learning of facts discourages transfer. Transfer is fostered when general principles of reasoning are taught together with self-monitoring practices and potential applications in varied contexts. Training in reasoning and critical thinking is only effective for transfer, when abstract principles and rules are coupled with examples. Transfer is promoted when learning takes place in a social context, which fosters generation of principles and explanations. Transfer improves when learning is through co-operative methods, and where there is feedback on performance with training examples. The specificity of the context in which principles are learned reduces their transfer. Transfer is promoted if learners are shown how problems resemble each other, if they are expected to learn to do this themselves, if they are aware of how to apply skills in different contexts, if attention is directed to the underlying goal structure of comparable problems, if examples are varied and are accompanied by rules or principles (especially if discovered by the learners), and if learners’ self-explanations are stimulated. Learning to use meta-cognitive strategies is especially important for transfer.     

Reasoning by analogy: Is it schema-mediated or case-based?

One of the “classical” ways of learning consists of studying examples of already solved problems. In two experiments, we analyzed the degree of abstraction of the knowledge used by ninth grade students to solve algebra problems after studying worked examples. The results showed that there are two processes underlying reasoning by analogy, one that uses abstract knowledge and another that involves case-based reasoning. Both experiments pointed out interindividual differences in the population under study: when given examples, some subjects seem to extract the structure of the solving process by comparing the worked examples, while others focus more on the specifics of each example. To these two processes correspond two levels of transfer: correctly solve problems that have the same structure as the examples, regardless of how similar they are, or be better at solving problems that resemble the examples the most. Experiment 2 used a dual-task paradigm to show that some subjects implement both processes, in which case the mental load is greater. This experiment also showed that both processes can lead to the long-term acquisition of the principles behind the examples.     

Coordinating principles and examples through analogy and self-explanation

Research on expertise suggests that a critical aspect of expert understanding is knowledge of the relations between domain principles and problem features. We investigated two instructional pathways hypothesized to facilitate students’ learning of these relations when studying worked examples. The first path is through self-explaining how worked examples instantiate domain principles and the second is through analogical comparison of worked examples. We compared both of these pathways to a third instructional path where students read worked examples and solved practice problems. Students in an introductory physics class were randomly assigned to one of three worked example conditions (reading, self-explanation, or analogy) when learning about rotational kinematics and then completed a set of problem solving and conceptual tests that measured near, intermediate, and far transfer. Students in the reading and self-explanation groups performed better than the analogy group on near transfer problems solved during the learning activities. However, this problem solving advantage was short lived as all three groups performed similarly on two intermediate transfer problems given at test. On the far transfer test, the self-explanation and analogy groups performed better than the reading group. These results are consistent with the idea that self-explanation and analogical comparison can facilitate conceptual learning without decrements to problem solving skills relative to a more traditional type of instruction in a classroom setting.     

Differential effects of problem-solving demands on individual and collaborative learning outcomes

The effectiveness and efficiency of individual versus collaborative learning was investigated as a function of instructional format among 140 high school students in the domain of biology. The instructional format either emphasized worked examples, which needed to be studied or the equivalent problems, which needed to be solved. Because problem solving imposes a higher cognitive load for novices than does studying worked examples it was hypothesized that learning by solving problems would lead to better learning outcomes (effectiveness) and be more efficient for collaborative learners, whereas learning by studying worked examples would lead to better learning outcomes and be more efficient for individual learners. The results supported these crossover interaction hypothesis. Consequences of the findings for the design of individual and collaborative learning environments are discussed.     

关于线性代数中的三类反问题研究

在线性代数课程的教与学中存在重视解决理论中的正面问题而忽略反问题的现象。针对这一现象,深入分析了线性方程组、特征值与特征向量、二次型这三类理论中的典型反问题,讨论了反问题的基本原理和求解方法,并给出了几个具体实例。     

Children's Reasoning as Collective Social Action through Problem Solving in Grade 2/3 Science Classrooms

Research on young children's reasoning show the complex relationships of knowledge, theories, and evidence in their decision-making and problem solving. Most of the research on children's reasoning skills has been done in individualized and formal research settings, not collective classroom environments where children often engage in learning and reasoning together to solve classroom problems. This study posits children's reasoning as a collective social activity that can occur in science classrooms. The study examined how children process their reasoning within the context of Grade 2/3 science classrooms and how the process of collectivity emerges from classroom interactions and dialogue between children as they attempt to solve their classroom problems. The study findings suggest that children's reasoning involves active evaluation of theories and evidence through collective problem solving, with consensus being developed through dialogical reasoning.     

Meaningful learning,reasoning ability,and students' understanding and problem solving of topics in genetics

The purpose of this study was to explore relationships among school students' (N = 189) meaningful learning orientation, reasoning ability and acquisition of meaningful understandings of genetics topics, and ability to solve genetics problems. This research first obtained measures of students' meaningful learning orientation (meaningful and rote) and reasoning ability (preformal and formal). Students were tested before and after laboratory-based learning cycle genetics instruction using a multiple choice assessment format and an open-ended assessment format (mental model). The assessment instruments were designed to measure students' interrelated understandings of genetics and their ability to solve and interpret problems using Punnett square diagrams. Regression analyses were conducted to examine the predictive influence of meaningful learning orientation, reasoning ability, and the interaction of these variables on students' performance on the different tests. Meaningful learning orientation best predicted students' understanding of genetics interrelationships, whereas reasoning ability best predicted their achievement in solving genetics problems. The interaction of meaningful learning orientation and reasoning ability did not significantly predict students' genetics understanding or problem solving. Meaningful learning orientation best predicted students' performance on all except one of the open-ended test questions. Examination of students' mental model explanations of meiosis, Punnett square diagrams, and relationships between meiosis and the use of Punnett square diagrams revealed unique patterns in students' understandings of these topics. This research provides information for educators on students' acquisition of meaningful understandings of genetics. © 1996 John Wiley & Sons, Inc.     

On the Concrete Nature of Human Thinking: content and context in analogical transfer

Although the development of abstract knowledge and critical thinking skills has often been extolled as an ideal and as the goal of education (Nisbett et al., 1987), much research in problem solving and other cognitive endeavours points to the role that concrete information and examples play in thinking (Medin & Ross, 1989; Reeves & Weisberg, in press). We discuss the concrete vs abstract knowledge debate in terms of the use of analogies in problem solving and conclude that exemplar‐specific details of problems and the context in which problems are learned guide the transfer of a solution principle from one base problem to a target. The content of problems—what they are about—is often more important than the more abstract, schematic solution principle in influencing retrieval of base analogues (Holyoak & Koh, 1987; Keane, 1987; Ross, 1987) and mapping of the solution principle from base to target (Gentner & Toupin, 1986; Ross, 1987, 1989; Fong & Nisbett, 1991). It is suggested that learning abstract solution principles in a domain (e.g. algebra, physics) benefits greatly when instruction is accompanied by examples illustrating those principles (Cheng et al., 1986; Fong et al., 1986) and that analogical transfer itself serves as a useful means to greater comprehension of a domain (Ross & Kennedy, 1990).     

Providing ability to probability: Reducing cognitive load through worked‐out examples

Undergraduate students tend to struggle with probability in their introductory statistics course. Probability problem solving requires several steps. First, students must make sense of the probability scenario, then determine the appropriate probability rules, and finally, execute the procedures to solve the problem. With no previous exposure to probability, this presents too great a cognitive load for many students. Using worked‐out problems then transitioning to partially worked‐out problems in an introductory statistics course at a large university helped students succeed at solving probability problems. The worked‐out problems included writing prompts to encourage self‐explanation of students' thinking through studying the worked‐out examples. This paper explains the use of these instructional principles and their implementation in an introductory statistics course for non‐STEM majors, resulting in higher student achievement and understanding.     

Developing Model‐Based Reasoning

Key elements of the structure and function of models in mathematics and science are identified. These elements are used as a basis for discussing the development of model‐based reasoning. A microgenetic study examines the beginnings of model‐based reasoning in a pair of fourth‐ and fifth‐grade children who solved several problems about chance and probability. Results are reported in the form of a cognitive model of children's problem‐solving performance. The cognitive model explains a transition in children's reasoning from tacit reliance on empirical regularity to a form of model‐based reasoning. Several factors fostering change in children's thinking are identified, including the role of notations, peer interaction, and teacher assistance. We suggest that model‐based reasoning is a slowly‐developing capability that emerges only with proper contextual and social support and that future study should be carried out in classrooms, where these forms of assistance can also be part of the object of study.

Model‐based reasoning is a significant intellectual milestone because it bridges the worlds of personal, intuitive knowledge, on the one hand, and mathematical‐scientific theory, on the other. However, across disciplines, consensus is still forming about what model‐based reasoning comprises, and there is little knowledge about its ontogenetic origins or how it develops. We consider analogy as the core of modeling, because in model‐based reasoning a system in one domain is used to understand a system in another. To understand how models come to play a role in reasoning, it is important to initiate study of their origins. Accordingly, we report a microgenetic study examining the beginnings of model‐based reasoning in a pair of young children solving problems about chance and probability. In this study we are engaged in the enterprise of modeling the development of modeling. That is, we report our results in the form of a cognitive model of children's problem‐solving performance that explains a transition in reasoning from a tacit reliance on empirical regularity to a form of model‐based reasoning. It is important to note the two distinct meanings for the term model used in this article. The first describes how children come to understand and appropriate a system of reasoning exemplified in practices of modeling. The second describes a research tool, a model of human reasoning—specifically, how children in this study began to use models of probability to reason about uncertain events. In this report, we use the terms model or model‐based reasoning to refer to the former interpretation, whereas references to a cognitive model denote the simulation of children's thinking—in this case, implemented as a computer program.

Before describing the empirical work, we first identify some key elements of the structure and function of models. Next, these elements of modeling are used as the basis for generating some conjectures about the development of model‐based reasoning. We describe a task that we used as a window to understanding progression in student reasoning toward reliance on models as tools for thought. We present our rationale for developing cognitive models of student performance and explain some choices concerning the implementation of the cognitive model reported here. Finally, we turn to the children's performance on chance and probability tasks and explain how that performance illuminates both what children do not understand about models and the kinds of relevant knowledge that they are acquiring.     

Process-Oriented Worked Examples: Improving Transfer Performance Through Enhanced Understanding

The research on worked examples has shown thatfor novices, studying worked examples is oftena more effective and efficient way of learningthan solving conventional problems. Thistheoretical paper argues that addingprocess-oriented information to worked examplescan further enhance transfer performance,especially for complex cognitive skills withmultiple possible solution paths.Process-oriented information refers to theprincipled (``why'') and strategic (``how'')information that experts use when solvingproblems. From a cognitive load perspective,studying the expert's ``why'' and ``how''information can be seen as constituting agermane cognitive load, which can fosterstudents' understanding of the principles of adomain and the rationale behind the selectedoperators, and their knowledge about howexperts select a strategy, respectively. Issueswith regard to the design, implementation, andassessment of effects of process-orientedworked examples are discussed, as well as thequestions they raise for future research.     

Open-ended approaches to science assessment using computers

We discuss the potential role of technology in evaluating learning outcomes in large-scale, widespread science assessments of the kind typically done at ETS, such as the GRE, or the College Board SAT II Subject Tests. We describe the current state-of-the-art in this area, as well as briefly outline the history of technology in large-scale science assessment and ponder possibilities for the future. We present examples from our own work in the domain of chemistry, in which we are designing problem solving interfaces and scoring programs for stoichiometric and other kinds of quantitative problem solving. We also present a new scientific reasoning item type that we are prototyping on the computer. It is our view that the technological infrastructure for large-scale constructed response science assessment is well on its way to being available, although many technical and practical hurdles remain.     

Effects of problem solving after worked example study on secondary school children’s monitoring accuracy

Monitoring accuracy, measured by judgements of learning (JOLs), has generally been found to be low to moderate, with students often displaying overconfidence, and JOLs of problem solving are no exception. Recently, primary school children’s overconfidence was shown to diminish when they practised problem solving after studying worked examples. The current study aimed to extend this research by investigating whether practising problem solving after worked example study would also improve JOL accuracy in secondary education. Adolescents of 14–15 years old (N = 143) were randomly assigned to one of five conditions that differed in timing of JOLs, whether practice problems were provided, and timing of the practice problems provided: (1) worked examples – JOL, (2) worked examples – delay – JOL, (3) worked examples – practice problems – JOL, (4) worked examples – practice problems – delay – JOL or (5) worked examples – delay – practice problems – JOLs. Results showed that practice problems improved absolute accuracy of JOLs as well as regulation accuracy. No differences in final test performance were found.     

A model for optimizing step size of learning tasks in competency-based multimedia practicals

Learners are often overwhelmed by the complexity of realistic learning tasks, but reducing this complexity through traditional Instructional Design (ID) methods jeopardizes the authenticity of the learning experience. To solve this apparent paradox, a two-phase ID model is presented. Phase 1 consists of cognitive task analysis, where a systematic approach to problem solving (SAP) is identified in conjunction with skill decomposition and determination of task complexity. In the subsequent design phase, inductive micro-level sequencing based on the four-component ID model (van Merriënboer, 1997) is applied where worked-out examples and problems accompanied by process worksheets assure the necessary variability of practice. Step size in a multiple-step whole-task approach—needed for the process worksheets—is determined on the basis of estimated part-task complexity. A developmental study of the model is illustrated with examples from the domain of law.     

Conceptual Understanding of Causal Reasoning in Physics

College students often experience difficulties in solving physics problems. These difficulties largely result from a lack of conceptual understanding of the topic. The processes of conceptual learning reflect the nature of the causal reasoning process. Two major causal reasoning methods are the covariational and the mechanism‐based approaches. This study was to investigate the effects of different causal reasoning methods on facilitating students’ conceptual understanding of physics. 125 college students from an introduction physics class were assigned into covariational group, mechanism‐based group, and control group. The results show that the mechanism‐based group significantly outperformed the other two groups in solving conceptual problems. However, no significant difference was found in all three groups performance on solving computational problems. Speculation on the inconsistent performance of the mechanism‐based group in conceptual and computational problem solving is given. Detailed analyses of the results, findings, and educational implications are discussed     

The Relative Merits of Explicit and Implicit Learning of Contrasted Algebra Principles

Knowledge representations that result from practicing problem solving can be expected to differ from knowledge representations that emerge from explicit verbalizing of principles and rules. We examined the degree to which the two types of learning improve problem-solving knowledge and verbal explanation knowledge in classroom instruction. We presented algebraic addition and multiplication problems to 153 sixth graders randomly assigned to two conditions. Students in the explicit learning condition had to verbally compare contrasted algebra problems. Students in the implicit learning condition had to generate and solve new problems. On three follow-up tests over 10 weeks, students in the explicit learning condition exhibited better problem-solving knowledge than students in the implicit learning condition, as well as some advantages in verbal concept knowledge. Implicit learning showed some advantages on not directly taught but incidentally learned aspects. Overall, this outcome favors the explicit learning of concepts. Explicit comparison fostered student performance on non-verbal and verbal measures, indicating that verbalization facilitates effective comparison.     

Learning From Examples: Fostering Self-Explanations in Computer-Based Learning Environments

Cognitive skills acquisition involves developing the ability to solve problems in knowledge-rich task domains, and is particularly important for any individual attempting to meet the challenges of our modern, knowledge-driven economy. This type of economy argues for reconceptualizing cognitive skills acquisition as a lifelong process. Research has shown that worked-out examples are the key to initial cognitive skill acquisition and, therefore, critical to lifelong learning. The extent to which learners' profit from the study of examples, however, depends on how well they explain the solutions of the examples to themselves. This paper discusses our own research on different types of computer-based learning environments that indirectly foster self-explanations by (a) fostering anticipative reasoning, (b) supporting self-explanations during the transition from example study to problem solving, and (c) supporting self-explanation activities with instructional explanations. It also discusses ways of leveraging new computer and video technologies to enhance these environments by representing problem situations and their concepts dynamically. The paper concludes by suggesting that these learning environments, if employed successfully, can encourage systematic, lifelong learning.