Cognitive Load During Problem Solving: Effects on Learning

Considerable evidence indicates that domain specific knowledge in the form of schemas is the primary factor distinguishing experts from novices in problem-solving skill. Evidence that conventional problem-solving activity is not effective in schema acquisition is also accumulating. It is suggested that a major reason for the ineffectiveness of problem solving as a learning device, is that the cognitive processes required by the two activities overlap insufficiently, and that conventional problem solving in the form of means-ends analysis requires a relatively large amount of cognitive processing capacity which is consequently unavailable for schema acquisition. A computational model and experimental evidence provide support for this contention. Theoretical and practical implications are discussed.     

Generative Activity During Geometry Problem Solving: Comparison of the Performance of High-Achieving and Low-Achieving High School Students

The problem-solving performance of groups of high-achieving (HA) and low-achieving (LA) Year 11 students was compared during solution of geometry problems using a think-aloud procedure. Students also undertook knowledge assessment tasks involving theorem recall and a prompted recall task to provide estimates of knowledge available for access during the solution attempts. Detailed analysis of problem-solving protocols indicated that HA students not only accessed a greater body of geometrical knowledge but also used that knowledge more effectively. On the more difficult problems, these HA students generated more information, used that generated information to access further relevant knowledge, and showed more frequent management of their processing behavior. The nature of the difference in generative activity of a student from each of the two groups is illustrated.     

Students' Coordination of Knowledge When Learning to Model Physical Situations

In this article, I present a study in which 12 pairs of 8th-grade students solved problems about a physical device with algebra. The device, called a winch, instantiates motions that can be modeled by pairs of simultaneous linear functions. The following question motivated the study: How can students generate algebraic models without direct instruction from more experienced others? The first main result of the study is that students have and can use criteria for judging when 1 algebraic expression is better than another. Thus, students can use criteria to regulate their problem-solving activity. The second main result is that constructing knowledge for modeling with algebra can require students to coordinate criteria for algebraic representations with several other types of knowledge that I also identify in the article. These results contribute to research on students' algebraic modeling, cognitive processes and knowledge structures for using mathematical representations, and the development of mathematical knowledge.     

样例和练习在促进解题迁移能力中的作用

通过一个2×2×2的因素实验,对96名初一学生在解题迁移能力中受样例和练习的影响进行了研究。结果表明,结合样例进行的练习促进了技能的熟练和解题能力的迁移;练习本身并不总能保证促进技能的熟练和解题能力的迁移,它至少要受三方面因素的影响第一,与在练习中是否有来自外部的指导和反馈有关;第二,与练习的任务性质有关;第三,与参与练习的个体智力和认知水平有关。     

Learning by imagining

Five experiments were conducted to investigate the relative effectiveness of 2 alternative instructional strategies. Students who were engaged in studying worked examples that emphasized understanding and remembering procedures and concepts were compared with students who were engaged in imagining worked examples that emphasized imagining procedures and concepts. It was hypothesized that students who held prerequisite schemas would find imagining to have a beneficial effect on learning, compared with studying the material, whereas students who were less knowledgeable would find imagining to have a negative effect on learning, compared with studying. Experimental results were in accord with our hypotheses. It was concluded that, under specific circumstances, encouraging students to imagine procedures and concepts can substantially facilitate learning.     

英语语法规则获得与迁移的智力技能研究

选取451名高中生为被试,以强调句型语法为研究内容,采取教师教学、产生式规则和专家思维编制的实验材料,探讨了英语语法规则获得早、中期阶段促进规则获得的手段。在前面实验的基础上,选取298名高中生,以定语从句为研究内容,运用产生式规则、专家思维和练习编制的不同学习材料,采取随机区组设计,探讨了英语语法规则获得后期阶段的智力技能研究。通过研究得到如下结论：在语法规则获得的早中期阶段,产生式样例有助于学生智力技能的获得,同时也有助于所获得技能的迁移;在智力技能获得的后期阶段,专家样例有利于学生形成专家头脑中的图式,有助于学生实现有关智力技能的自动化。     

Mistakes on display: Incorrect examples refine equation solving and algebraic feature knowledge

Although findings from cognitive science have suggested learning benefits of confronting errors (Metcalfe, 2017), they are not often capitalized on in many mathematics classrooms (Tulis, 2013). The current study assessed the effects of examples focused on either common mathematical misconceptions and errors or correct concepts and procedures on algebraic feature knowledge and solving quadratic equations. Middle school algebra students (N = 206) were randomly assigned to four conditions. Two errorful conditions either displayed errors and asked students to explain or displayed correct solutions and primed students to reflect on potential errors by problem type. A correct example condition and problem-solving control group were also included. Studying and explaining common errors displayed in incorrect examples improved equation-solving ability. An aptitude-by-treatment interaction revealed that learners with limited understandings of algebraic features demonstrated greater benefits. Theoretical implications about using examples to promote learning from errors are considered in addition to suggestions for educational practice.     

Does calling it “Morgan's way” reduce student learning? Evaluating the effect of person-presentation during comparison and discussion of worked examples in mathematics classrooms

Mathematics textbooks sometimes present worked examples as being generated by particular fictitious students (i.e., person-presentation). However, there are indicators that person-presentation of worked examples may harm generalization of the presented strategies to new problems. In the context of comparing and discussing worked examples during extended classroom instruction, the current study compared the impact of person-presentation to strategy labels on students' posttest accuracy and ratings of strategy generalizability. Five algebra teachers and their 168 students used worked examples either presented using fictitious students or with a strategy label during a multiweek unit on equation solving, with teachers randomly assigned to condition. All students compared and discussed the worked examples. In this context, we found no effect of condition on student accuracy at posttest, nor on their ratings of the generalizability of the presented strategies. We discuss why previously found negative effects of person-presentation may not have extended to this context.     

The effects of cumulative practice on mathematics problem solving

This study compared three different methods of teaching five basic algebra rules to college students. All methods used the same procedures to teach the rules and included four 50-question review sessions interspersed among the training of the individual rules. The differences among methods involved the kinds of practice provided during the four review sessions. Participants who received cumulative practice answered 50 questions covering a mix of the rules learned prior to each review session. Participants who received a simple review answered 50 questions on one previously trained rule. Participants who received extra practice answered 50 extra questions on the rule they had just learned. Tests administered after each review included new questions for applying each rule (application items) and problems that required novel combinations of the rules (problem-solving items). On the final test, the cumulative group outscored the other groups on application and problem-solving items. In addition, the cumulative group solved the problem-solving items significantly faster than the other groups. These results suggest that cumulative practice of component skills is an effective method of training problem solving.     

Arithmetic word problem solving: a Situation Strategy First framework

Before instruction, children solve many arithmetic word problems with informal strategies based on the situation described in the problem. A Situation Strategy First framework is introduced that posits that initial representation of the problem activates a situation-based strategy even after instruction: only when it is not efficient for providing the numerical solution is the representation of the problem modified so that the relevant arithmetic knowledge might be used. Three experiments were conducted with Year 3 and Year 4 children. Subtraction, multiplication and division problems were created in two versions involving the same wording but different numerical values. The first version could be mentally solved with a Situation strategy (Si version) and the second with a Mental Arithmetic strategy (MA version). Results show that Si-problems are easier than MA-problems even after instruction, and, when children were asked to report their strategy by writing a number sentence, equations that directly model the situation were predominant for Si-problems but not for MA ones. Implications of the Situation Strategy First framework regarding the relation between conceptual and procedural knowledge and the development of arithmetic knowledge are discussed.     

Providing worked examples for learning multiple principles

Worked examples support learning. However, if they introduce easy-to-confuse concepts or principles, specific ways of providing worked examples may influence their effectiveness. Multiple worked examples can be introduced blocked (i.e., several for the same principle) or interleaved (i.e., switching between principles), and can be sequentially or simultaneously presented. Crossing these two factors provides four ways of presenting worked examples: blocked/sequential, interleaved/sequential, blocked/simultaneous, and interleaved/simultaneous. In an experiment with university students (N = 174), we investigated how these two factors influence the acquisition of procedural and conceptual knowledge about different, but closely related (thus, easy-to-confuse) stochastic principles. Additionally, we assessed the ability of students to discriminate between principles with verification tasks. Simultaneous presentation benefitted procedural knowledge whereas, interleaved presentation benefitted conceptual knowledge. No significant differences were found for verification tasks. The results suggest that it is worthwhile to adapt the presentation of the worked examples to the learning goals.     

Abstract Planning and Perceptual Chunks: Elements of Expertise in Geometry

We present a new model of skilled performance in geometry proof problem solving called the Diagram Configuration model (DC). While previous models plan proofs in a step-by-step fashion, we observed that experts plan at a more abstract level: They focus on the key steps and skip the less important ones. DC models this abstract planning behavior by parsing geometry problem diagrams into perceptual chunks, called diagram configurations, which cue relevant schematic knowledge. We provide verbal protocol evidence that DC's schemas correspond with the step-skipping inferences experts make in their initial planning. We compare DC with other models of geometry expertise and then, in the final section, we discuss more general implications of our research. DC's reasoning has important similarities with Larkin's (1988) display-based reasoning approach and Johnson-Laird's (1983) mental model approach. DC's perceptually based schemas are a step towards a unified explanation of (1) experts' superior problem-solving effectiveness, (2) experts' superior problem-state memory, and (3) experts' ability, in certain domains, to solve relatively simple problems by pure forward inferencing. We also argue that the particular and efficient knowledge organization of DC challenges current theories of skill acquisition as it presents an end-state of learning that is difficult to explain within such theories. Finally, we discuss the implications of DC for geometry instruction.     

小学生代数运算规则的样例学习

采用完整或不完整样例,对180名六年级小学生用样例学习两种代数运算规则进行了实验研究。结果显示：多数被试难以学会“平方差”运算规则,只有少数被试学会了“完全平方和”运算规则;反馈对不完整样例的学习效果有促进作用;用不完整样例学习难度不同的规则,其效果不同;在无反馈条件下,完整样例的学习效果都好于不完整样例;在有反馈的情况下,只有删除一步运算步骤样例的学习效果比完整样例的学习效果好     

Eliciting Self-Explanations Improves Understanding

Learning involves the integration of new information into existing knowledge. Generating explanations to oneself (self-explaining) facilitates that integration process. Previously, self-explanation has been shown to improve the acquisition of problem-solving skills when studying worked-out examples. This study extends that finding, showing that self-explanation can also be facilitative when it is explicitly promoted, in the context of learning declarative knowledge from an expository text. Without any extensive training, 14 eighth-grade students were merely asked to self-explain after reading each line of a passage on the human circulatory system. Ten students in the control group read the same text twice, but were not prompted to self-explain. All of the students were tested for their circulatory system knowledge before and after reading the text. The prompted group had a greater gain from the pretest to the posttest. Moreover, prompted students who generated a large number of self-explanations (the high explainers) learned with greater understanding than low explainers. Understanding was assessed by answering very complex questions and inducing the function of a component when it was only implicitly stated. Understanding was further captured by a mental model analysis of the self-explanation protocols. High explainers all achieved the correct mental model of the circulatory system, whereas many of the unprompted students as well as the low explainers did not. Three processing characteristics of self-explaining are considered as reasons for the gains in deeper understanding.     

Individual differences in the preference for worked examples: Lessons from an application of dispositional learning analytics

Worked-examples have been established as an effective instructional format in problem-solving practices. However, less is known about variations in the use of worked examples across individuals at different stages in their learning process in student-centred learning contexts. This study investigates different profiles of students' learning behaviours based on clustering learning dispositions, prior knowledge, and the choice of feedback strategies in a naturalistic setting. The study was conducted on 1,072 students over an 8-week long introductory mathematics course in a blended instructional format. While practising exercises in a digital learning environment, students can opt for tutored problem solving, untutored problem solving, or call worked examples. The results indicated six distinct profiles of learners regarding their feedback preferences in different learning phases. Finally, we investigated antecedents and consequences of these profiles and investigated the adequacy of used feedback strategies concerning ‘help-abuse’. This research indicates that the use of instructional scaffolds as worked-examples or hints and the efficiency of that use differs from student to student, making the attempt to find patterns at an overall level a hazardous endeavour.     

The efficiency of multimedia learning into old age

Background: On the basis of a multimodal model of working memory, cognitive load theory predicts that a multimedia‐based instructional format leads to a better acquisition of complex subject matter than a purely visual instructional format. Aims: This study investigated the extent to which age and instructional format had an impact on training efficiency among both young and old adults. It was hypothesised that studying worked examples that are presented as a narrated animation (multimedia condition) is a more efficient means of complex skill training than studying visually presented worked examples (unimodal condition) and solving conventional problems. Furthermore, it was hypothesised that multimedia‐based worked examples are especially helpful for elderly learners, who have to deal with a general decline of working‐memory resources, because they address both mode‐specific working‐memory stores. Sample: The sample consisted of 60 young (mean age = 15.98 years) and 60 old adults (mean age = 64.48 years). Methods: Participants of both age groups were trained in either a conventional, a unimodal, or a multimedia condition. Subsequently, they had to solve a series of test problems. Dependent variables were perceived cognitive load during the training, performance on the test, and efficiency in terms of the ratio between these two variables. Results: Results showed that for both age groups multimedia‐based worked examples were more efficient than the other training formats in that less cognitive load led to at least an equal performance level. Conclusion: Although no difference in the beneficial effect of multimedia learning was found between the age groups, multimedia‐based instructions seem promising for the elderly.     

Assessing complex problem-solving skills with multiple complex systems

In this paper we propose the multiple complex systems (MCS) approach for assessing domain-general complex problem-solving (CPS) skills and its processes knowledge acquisition and knowledge application. After defining the construct and the formal frameworks for describing complex problems, we emphasise some of the measurement issues inherent in assessing CPS skills with single tasks (i.e., fixed item difficulty, low or unknown reliability, and a large impact of random errors). With examples of the MicroDYN test and the MicroFIN test (two instances of the MCS approach), we show how to adequately score problem-solving skills by using multiple tasks. We discuss implications for problem-solving research and the assessment of CPS skills in general.