Developing procedural flexibility: Are novices prepared to learn from comparing procedures?

Background. A key learning outcome in problem‐solving domains is the development of procedural flexibility, where learners know multiple procedures and use them appropriately to solve a range of problems (e.g., Verschaffel, Luwel, Torbeyns, & Van Dooren, 2009 ). However, students often fail to become flexible problem solvers in mathematics. To support flexibility, teaching standards in many countries recommend that students be exposed to multiple procedures early in instruction and be encouraged to compare them. Aims. We experimentally evaluated this recommended instructional practice for supporting procedural flexibility during a classroom lesson, relative to two alternative conditions. The alternatives reflected the common instructional practice of delayed exposure to multiple procedures, either with or without comparison of procedures. Sample. Grade 8 students from two public schools (N= 198) were randomly assigned to condition. Students had not received prior instruction on multi‐step equation solving, which was the topic of our lessons. Method. Students learned about multi‐step equation solving under one of three conditions in math class for about 3 hr. They also completed a pre‐test, post‐test, and 1‐month‐retention test on their procedural knowledge, procedural flexibility, and conceptual knowledge of equation solving. Results. Novices who compared procedures immediately were more flexible problem solvers than those who did not, even on a 1‐month retention test. Although condition had limited direct impact on conceptual and procedural knowledge, greater flexibility was associated with greater knowledge of both types. Conclusions. Comparing procedures can support flexibility in novices and early introduction to multiple procedures may be one important reason.     

Moving to Learn: How Guiding the Hands Can Set the Stage for Learning

Previous work has found that guiding problem‐solvers' movements can have an immediate effect on their ability to solve a problem. Here we explore these processes in a learning paradigm. We ask whether guiding a learner's movements can have a delayed effect on learning, setting the stage for change that comes about only after instruction. Children were taught movements that were either relevant or irrelevant to solving mathematical equivalence problems and were told to produce the movements on a series of problems before they received instruction in mathematical equivalence. Children in the relevant movement condition improved after instruction significantly more than children in the irrelevant movement condition, despite the fact that the children showed no improvement in their understanding of mathematical equivalence on a ratings task or on a paper‐and‐pencil test taken immediately after the movements but before instruction. Movements of the body can thus be used to sow the seeds of conceptual change. But those seeds do not necessarily come to fruition until after the learner has received explicit instruction in the concept, suggesting a “sleeper effect” of gesture on learning.     

Do examples of failure effectively prepare students for learning from subsequent instruction?

Studies on the productive failure (PF) approach have demonstrated that attempting to solve a problem prepares students more effectively for later instruction compared to observing failed problem-solving attempts prior to instruction. However, the examples of failure used in these studies did not display the problem-solving-and-failing process, which may have limited the preparatory effects. In this quasi-experiment, we investigated whether observing someone else engaging in problem solving can prepare students for instruction, and whether examples that show the problem-solving-and -failing process are more effective than those that only show the outcome of this process. We also explored whether the perceived model–observer similarity had an impact on the effectiveness of observing examples of failure. The results showed that observing examples effectively prepares students for learning from instruction. However, observing the model's problem-solving-and-failing process did not prepare students more effectively than merely looking at the outcome. Studying examples were more effective if model–observer similarity was high.     

Learning new problem-solving strategies leads to changes in problem representation

Children sometimes solve problems incorrectly because they fail to represent key features of the problems. One potential source of improvements in children's problem representations is learning new problem-solving strategies. Ninety-one 3rd- and 4th-grade students solved mathematical equivalence problems (e.g., 3 + 4 + 6 = 3 + __) and completed a representation assessment in which they briefly viewed similar problems and either reconstructed each problem or identified it in a set of alternatives. Experimental groups then received a lesson about one or both of two solution strategies, the equalize strategy and the add–subtract strategy. A control group received no instruction. All children completed posttest assessments of representation and problem solving. Children taught the equalize strategy improved their problem representations more than those not taught it. This pattern did not hold for the add–subtract strategy. These results indicate that learning new strategies is one source of changes in problem representation. However, some strategies are more effective than others at promoting accurate problem representation.     

样例关键特征变异数量对学习解方程的影响

186名初一年级学生通过对比三种不同变异类型的样例学习解一元一次方程,他们对比方程问题类型和解法这两个关键特征的变异、只对比问题类型的变异、或者只对比解法的变异。结果显示样例关键特征不同变异类型的学习效果受到学生先前知识的影响：对于在前测中没有使用简便方法的学生,变异类型和解法两个关键特征比只变异类型或解法其中一个关键特征更有利于他们学习变通性知识和概念性知识;而对于在前测中有使用简便方法的学生,不同变异类型的效果没有显著差别。多重样例变异性的设计需要提供机会让学生充分对比学习每个关键特征。     

Enhancing Digital Fluency through a Training Program for Creative Problem Solving Using Computer Programming

The purpose of this paper is to propose a training program for creative problem solving based on computer programming. The proposed program will encourage students to solve real‐life problems through a creative thinking spiral related to cognitive skills with computer programming. With the goal of enhancing digital fluency through this proposed training program, we investigated its effects. Two sets of experiments were performed in which 119 typical students and 30 younger, gifted students participated. Two synthetic creative problem solving tests, which had a high correlation with logical ability, scientific problem solving ability and divergent thinking ability, were developed to measure creative problem solving ability. We provided the treatment group with a paper‐based booklet with relevant problems developed specifically for that group. ANCOVA statistical procedures were used to analyze the pre‐ and post‐synthetic creative problem solving tests. The findings of our study are as follows: with typical students, the originality of the treatment group outperformed the control group, a result that was compatible with previous research. With gifted students, the fluency of the treatment group outperformed the control group, and overall creative problem solving ability was enhanced. Remarkably, fluency increased significantly, a notable difference from the results of prior studies. In conclusion, we inferred that, given the definition of digital fluency, if creative problem solving ability is enhanced by a training program for creative problem solving based on computer programming, digital fluency will ultimately be improved. In this paper, we discuss the result of fluency enhancement that contradicts prior research. We suggest that this training program could be a new learning environment for the students who have grown up with digital media.     

Verbalization of discovered principles for solving match-stick problems

Abstract.— Problem solving performance of subjects who were taught part of a principle was studied under three experimental conditions, differing with respect to the amount of hints given as to the nature of the missing part of the principle, and compared with performance of suhjects who were taught the full principle. After practice, those who were taught the principle partially were asked to state the missing part of the principle, or any other general method found for solving the problems. Subjects who were taught the full principle solved most problems, while performance under experimental conditions varied with the preciseness with which the stated principle specified relevant parts of the problems to be solved. The hints failed to influence performance with respect to both problem solving and ability to state principles.     

Teaching students to solve insight problems: Evidence for domain specificity in creativity training

The purpose of this research was to investigate whether insight problem solving depends on domain‐specific or domain‐general problem‐solving skills, that is, whether people think in terms of conceptually different types of insight problems. In Study 1, participants sorted insight problems into categories. A cluster analysis revealed 4 main categories of insight problems: verbal, mathematical, spatial, and a combination of verbal or spatial. In Studies 2 and 3, participants received training in how to solve verbal, spatial, or mathematical problems, or all 3 types. They were taught that solutions to verbal insight problems lie in defining and analyzing the terms in the problem, solutions to mathematical insight problems lie in a novel approach to numbers, or solutions to spatial insight problems lie in removing a self‐imposed constraint. In both studies, the spatial trained group performed better than the verbal trained group on spatial problems but not on other types of problems. These findings are consistent with the idea that people mentally separate insight problems into distinct types. Implications for instruction in insight problem solving are discussed.     

学习困难儿童的问题解决特点研究

本研究选取学习困难儿童和正常儿童各 3 2名 ,设置河内塔问题解决的情境 ,采用临床观察法 ,对学习困难儿童在问题解决中的特点作了初步的探究。研究发现 :1与正常儿童相比 ,学习困难儿童在发现和有效运用策略方面明显不足 ,但当学习困难儿童对问题情境比较熟悉后 ,有明显的进步 ;2一定的提示并不能帮助学习困难儿童最有效地应用策略。     

Overcoming Instructor‐Originated Math Anxiety in Philosophy Students: A Consideration of Proven Techniques for Students Taking Formal Logic

Every university student has his or her nemesis. Biology and social science students anticipate with great apprehension their required statistics course, while many philosophy students live in fear of formal logic. Math anxiety is the common thread uniting all of them. This article argues that since formal logic is an algebra requiring similar kinds of symbol‐manipulation skills needed to succeed in a basic mathematics course, then if logic students have math anxiety, this can impede their progress. Further, it argues that math anxiety is primarily caused by and exacerbated by poor instruction. Formal logic instructors who employ effective instructional techniques for reducing it can help their students overcome math anxiety to foster learning. Methods of instruction leading to anxiety reduction and evidence supporting their efficacy are discussed, including co‐operative learning, the mastery goal approach, and self‐paced learning. None of these methods holds back more advanced students.     

Effects of peer tutoring and consequences on the math performance of elementary classroom students

The effects of unstructured peer-tutoring procedures on the math performance of fourth- and fifth-grade students were investigated. Students' performances in two daily math sessions, during which they worked problems of the same type and difficulty, were compared. When students tutored each other over the same math problems as they subsequently worked, higher accuracies and rates of performance were associated with the tutored math sessions. The use of consequences for accurate performance seemed to enhance the effects of tutoring on accuracy. The results from an independent-study control condition, which was the same peer-tutoring except that students did not interact with each other, suggested that interactions between students during the tutoring procedure were, in part, responsible for improved accuracy and rate of performance. When students tutored each other over different but related problems to those that they were subsequently asked to solve, accuracies and rates during tutored math sessions were also higher, suggesting the development of generalized skills in solving particular types of math problems.     

Iterating between lessons on concepts and procedures can improve mathematics knowledge

Background Knowledge of concepts and procedures seems to develop in an iterative fashion, with increases in one type of knowledge leading to increases in the other type of knowledge. This suggests that iterating between lessons on concepts and procedures may improve learning. Aims The purpose of the current study was to evaluate the instructional benefits of an iterative lesson sequence compared to a concepts‐before‐procedures sequence for students learning decimal place‐value concepts and arithmetic procedures. Samples In two classroom experiments, sixth‐grade students from two schools participated (N = 77 and 26). Method Students completed six decimal lessons on an intelligent‐tutoring systems. In the iterative condition, lessons cycled between concept and procedure lessons. In the concepts‐first condition, all concept lessons were presented before introducing the procedure lessons. Results In both experiments, students in the iterative condition gained more knowledge of arithmetic procedures, including ability to transfer the procedures to problems with novel features. Knowledge of concepts was fairly comparable across conditions. Finally, pre‐test knowledge of one type predicted gains in knowledge of the other type across experiments. Conclusions An iterative sequencing of lessons seems to facilitate learning and transfer, particularly of mathematical procedures. The findings support an iterative perspective for the development of knowledge of concepts and procedures.     

Learning from others in 9‐18‐month‐old infants

The use of an adult as a resource for help and instruction in a problem solving situation was examined in 9, 14, and 18‐month‐old infants. Infants were placed in various situations ranging from a simple means‐end task where a toy was placed beyond infants' prehensile space on a mat, to instances where an attractive toy was placed inside closed transparent boxes that were more or less difficult for the child to open. The experimenter gave hints and modelled the solution each time the infant made a request (pointing, reaching, or showing a box to the experimenter), or if the infant was unable to solve the problem. Infants' success on the problems, sensitivity to the experimenter's modelling, and communicative gestures (requests, co‐occurrence of looking behaviour and requests) were analysed. Results show that older infants had better success in solving problems although they exhibited difficulties in solving the simple means‐end task compared to the younger infants. Moreover, 14‐ and 18‐month‐olds were sensitive to the experimenter's modelling and used her demonstration cues to solve problems. By contrast, 9‐month‐olds did not show such sensitivity. Finally, 9‐month‐old infants displayed significantly fewer communicative gestures toward the adult compared to the other age groups, although in general, all infants tended to increase their frequency of requests as a function of problem difficulty. These observations support the idea that during the first half of the second year infants develop a new collaborative stance toward others. The stance is interpreted as foundational to teaching and instruction, two mechanisms of social learning that are sometime considered as specifically human. Copyright © 2006 John Wiley & Sons, Ltd.     

Reasoning and Instruction in Medical Curricula

We describe two experiments that examine the knowledge and explanatory processes of students in two medical schools with different modes of instruction. One school had a conventional curriculum with basic science courses taught 1 '/2 years before the clinical training; the other had a problem-based learning curriculum with basic science taught in the context of clinical problems and general problem-solving strategies involving knowledge elaboration and hypothetico-deductive reasoning. Both before and after being exposed to relevant basic science information, students were asked to provide diagnostic explanations of a clinical case. In this study, students in the problem-based learning curriculum reasoned in a manner consistent with the way they were taught, using a backward directed pattern of reasoning and extensive elaborations based on detailed biomedical information. However; these students had a greater tendency to commit errors of scientific fact, to generate less coherent explanations, and to use flawed patterns of explanation, such as circular reasoning. These results are viewed as reflecting the operation of two factors: context and method of instruction. The interaction between these factors is expressed in terms of the hypothesis that basic science and clinical knowledge constitute two different worlds.     

A visualization technique for Bayesian reasoning

We tested a method for solving Bayesian reasoning problems in terms of spatial relations as opposed to mathematical equations. Participants completed Bayesian problems in which they were given a prior probability and two conditional probabilities and were asked to report the posterior odds. After a pretraining phase in which participants completed problems with no instruction or external support, participants watched a video describing a visualization technique that used the length of bars to represent the probabilities provided in the problem. Participants then completed more problems with a chance to implement the technique by clicking interactive bars on the computer screen. Performance improved dramatically from the pretraining phase to the interactive‐bar phase. Participants maintained improved performance in transfer phases in which they were required to implement the visualization technique with either pencil‐and‐paper or no external medium. Accuracy levels for participants using the visualization technique were very similar to participants trained to solve the Bayes theorem equation. The results showed no evidence of learning across problems in the pretraining phase or for control participants who did not receive training, so the improved performance of participants using the visualization method could be uniquely attributed to the method itself. A classroom sample demonstrated that these benefits extend to instructional settings. The results show that people can quickly learn to perform Bayesian reasoning without using mathematical equations. We discuss ways that a spatial solution method can enhance classroom instruction on Bayesian inference and help students apply Bayesian reasoning in everyday settings.     

How learning contexts facilitate strategy transfer

The present study assessed the role of context in the acquisition and transfer of a mathematical strategy. One hundred and six children were assigned to four conditions: direct strategy instruction, guided discovery, direct teaching plus discovery, or a control condition. The intervention consisted of fourteen sessions during which the number-family strategy, useful for addition and subtraction, was taught. Third grade students in the guided discovery condition performed better than those in the direct instruction condition on far transfer problems that measured deep conceptual understanding. Students who had total or partial exposure to guided discovery held stronger beliefs and adopted more positive goals about the importance of mathematical understanding and peer collaboration, attributed less importance to task extrinsic reasons for success, and reported greater use of deep processing strategies than students exposed to direct, explicit instructions. Finally, students in the discovery conditions were able to communicate more effectively during problem solving than students in the direct instructions condition.     

Comparison of accommodations and interventions for youth with ADHD: A randomized controlled trial

School psychologists have a variety of evidence-based interventions from which to choose when recommending classroom-based strategies for students with attention deficit hyperactivity disorder (ADHD); however, strategies frequently found on individualized education plans are accommodations designed to remove barriers to learning, which have limited empirical evidence. As such, the purpose of the current study was to compare the efficacy of three interventions (i.e., organization training, self-management, note-taking instruction) and three accommodations (i.e., organization support, extended time, copy of teacher notes) to address difficulties with organization and maintaining attention during a science lesson and associated independent practice. The study included 64 middle school students with ADHD randomized to either an intervention or an accommodation condition. The intervention group was further divided into two subgroups, consisting of (a) students who were willing to follow intervention procedures and (b) students who were not willing to follow the procedures (behavioral indicators of social validity). Results indicated that adolescents with ADHD in the intervention group were statistically significantly more likely to organize and maintain binder organization and to take complete and accurate notes than those in the accommodation group. In addition, exploratory analyses indicated that adolescents who demonstrated willingness to follow intervention procedures were more likely to be academically engaged during instruction and independent work and to complete independent work accurately than those who resisted the procedures.     

样例类型与解释方式对初中生数学概率问题解决的效果

为考察样例类型与解释方式对初中生数学概率问题解决的促进作用,实验1随机选取初中生90名,比较正确样例组、正误样例组、对照组的学习效果,实验2随机选取另外90名初中生,比较有教学解释、有自我解释与无解释的正误样例组的即时与延时测试学习效果,研究发现:(1)正误样例学习效果显著好于正确样例;(2)有解释的正误样例学习效果显著好于无解释的正误样例;(3)与有教学解释的正误样例学习效果相比,有自我解释的正误样例学习效果显著且更持久。     

USING NUMBER LINE PROCEDURES AND PEER TUTORING TO IMPROVE THE MATHEMATICS COMPUTATION OF LOW-PERFORMING FIRST GRADERS

Recommendations for mathematics instruction frequently include the use of manipulatives as a critical component. There are few experimental analyses of teaching strategies involving the use of manipulatives (e.g., the number line). This investigation used a multiple baseline design across three groups of students to examine the effectiveness of an experimental procedure for improving low-performing children's skills in solving missing addend arithmetic problems using the number line. To address concerns about inadequate time for instruction and insufficient practice in most mathematics instruction, trained peer tutors implemented the procedure. The results suggest that student performance improved when trained tutors taught the students number line procedures and gave them feedback on accuracy. Further, social validation data indicate that the students, their tutors, and their classroom teachers liked the procedures.     

Knowledge in transition: Adults' developing understanding of a principle of physical causality

What are the conditions that make it likely that cognitive change will occur? We investigate this issue with respect to 25 college students' developing understanding of gear movement (a particular problem in the domain of physical causation). The participants solved problems, then received minimal instruction, and solved additional problems. Significantly, only some of the participants changed their approach to solving the problems after receiving instruction; the remainder of the participants were stable in their understanding and either continued to solve all problems correctly or continued to solve key problems incorrectly. Most analyses focused on the participants who began by solving problems incorrectly. In particular, we attempted to differentiate those participants who exhibited cognitive change from those who did not. To do this, we examined precursors of knowledge change that were motivated by different theoretical positions on mechanisms of cognitive change and development (i.e., consideration of multiple approaches, cognitive conflict, and instruction as an example of a sociocultural process). Results suggest that having multiple approaches available and using instructional information to build on not-well-developed conceptions are likely candidates for understanding knowledge change for adult participants with respect to their developing understanding of physical causality.