Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students' errors     

Dirk De Bock  Wim Van Dooren  Dirk Janssens  Lieven Verschaffel 《Educational Studies in Mathematics》2002,50(3):311-334

Several recent ascertaining studies revealed a deep-rooted and almost irresistible tendency among 12–16-year old students to improperly apply the linear or proportional model in word problems involving lengths, areas and volumes of similar plane figures and solids. While these previous studies showed to what extent students' improper use of linear reasoning is affected by different characteristics of the task, it remained largely unclear what aspects of their knowledge base are responsible for the occurrence and strength of this phenomenon and how these aspects relate to other more general misconceptions and buggy rules identified in the literature. This paper reports an in-depth investigation by means of individual semi-standardised interviews aimed at analysing the thinking process underlying students' improper linear reasoning and how this process is affected by their mathematical conceptions, beliefs and habits. During these interviews,students' solution processes were revealed through a number of well-specified questions by the interviewer with respect to one single non-linear application problem, as well as through their reactions to subsequent kinds of cognitive conflict. The interviews provided a lot of information about the actual process of problem solving from students falling into the ‘linearity trap’ and the mechanism behind it. Although some students seem to really ‘believe’ that quantities are always linked proportionally, their improper use of linearity often results from superficial and intuitive reasoning, influenced by specific mathematical conceptions, habits and beliefs leading to a deficient modelling process. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

Strategic competence: Applying Siegler’s theoretical and methodological framework to the domain of simple addition     

Joke Torbeyns  Lieven Verschaffel  Pol Ghesquière 《European Journal of Psychology of Education - EJPE》2002,17(3):275-291

In this study we investigated the variability, frequency, efficiency, and adaptiveness of young children’s strategy use in the domain of simple addition by means of the choice/no-choice method. Seventy-seven beginning second-graders, divided in 3 groups according to general mathematical ability, solved a series of 25 simple additions in 3 different conditions. In the first condition, children could choose whatever strategy they wanted to solve each problem. In the second and third condition, the same children had to solve all problems with one particular strategy, respectively adding up to 10 and retrieval. The results demonstrate that second-graders as a whole choose adaptively between retrieval, decomposition, and counting strategies when solving simple additions, and that they use these strategies neither equally frequently nor equally efficiently. Furthermore, our results indicate that children with different mathematical ability use generally the same strategies to solve these problems, but differ in the frequency, accuracy and adaptiveness with which they apply these strategies. Finally, this study documents the value of the choice/no-choice method to assess the adaptiveness of young children’s strategy use in the domain of early arithmetic.  相似文献   

The CLIA-model: A framework for designing powerful learning environments for thinking and problem solving     

Erik De Corte  Lieven Verschaffel  Chris Masui 《European Journal of Psychology of Education - EJPE》2004,19(4):365-384

A major challenge for education and educational research is to build on our present understanding of learning for designing environments for education that are conducive to fostering in students self-regulatory and cooperative learning skills, transferable knowledge, and a disposition toward competent thinking and problem solving. Taking into account inquiry-based knowledge on learning and recent instructional research, this article presents the CLIA-model (Competence, Learning, Intervention, Assessment) as a framework for the design of learning environments aimed to be powerful in eliciting in students learning processes that facilitate the acquisition of productive knowledge and competent learning and thinking skills. Next, two intervention studies are described that embody major components of this framework, one focussing on mathematical problem solving in primary school, and a second one relating to self-regulatory skills in university freshmen. Both studies were carried out in parallel with the development of the framework, and were instrumental in identifying and specifying the different components of the model. They yielded both promising initial support for the model by showing that CLIA-based learning environments are indeed powerful in facilitating in students the acquisition of high-literacy learning results, especially the acquisition and transfer of self-regulation skills for learning and problem solving.  相似文献   

如何构建有效的学习环境--欧洲第十一届学习与教学研究大会综述     

陈丽敏  陈琦 《数学教育学报》2006,15(1):57-59

2005年8月23~27日,欧洲第十一届关于学习与教学研究的大会在塞浦路斯首都尼科西亚举行.这次大会探讨了当前教育研究中的两个方面:一方面是我们需要通过多种教学途径来加强学生的知识学习过程;另一方面,当前的教育研究要充分认识到有效教学情境的创设在学习与教学中的重要作用.一些教育专家从社会情感,学习与认知理论,教育评价等几个方面对有效学习环境的构建做了专题报告.作为教育研究的一个重要领域的数学教育在此次大会上是一个很重要的专题,数学教育专家的一些观点和研究成果对于我国的数学课程与教学改革具有深刻的启示.  相似文献   

AN INVESTIGATION ON CHINESE TEACHERS�� REALISTIC PROBLEM POSING AND PROBLEM SOLVING ABILITY AND BELIEFS     

Limin Chen  Wim Van Dooren  Qi Chen  Lieven Verschaffel 《International Journal of Science and Mathematics Education》2011,9(4):919-948

In the present study, which is a part of a research project about realistic word problem solving and problem posing in Chinese elementary schools, a problem solving and a problem posing test were administered to 128 pre-service and in-service elementary school teachers from Tianjin City in China, wherein the teachers were asked to solve 3 contextually challenging division-with-remainder (DWR) word problems and pose word problems according to 3 symbolic expressions. Afterwards, they were also given 2 questionnaires wherein they had to evaluate 3 different pupil reactions to, respectively, 1 problem solving item and 1 problem posing item about DWR. First, our results revealed that teachers behaved quite ‘realistically’ not only when solving and posing DWR problems themselves but also when evaluating elementary school pupils’ DWR problem solving and problem posing performance. Second, we found a correspondence between teachers’ own performance on the tests and their evaluations of pupils’ reactions. Third, the present study provides some further insight into the role of one of the instructional factors that is generally considered responsible for the strong and worldwide tendency among elementary school children to neglect real-world knowledge and realistic considerations in their endeavours to solve and pose mathematical word problems, namely the teachers’ conceptions and beliefs about this topic.  相似文献   

KNOWLEDGE ON ACCELERATED MOTION AS MEASURED BY IMPLICIT AND EXPLICIT TASKS IN 5 TO 16 YEAR OLDS     

Mirjam Ebersbach  Wim Van Dooren  Lieven Verschaffel 《International Journal of Science and Mathematics Education》2011,9(1):25-46

The present study aimed at investigating children's and adolescents' understanding of constant and accelerated motions. The main objectives were (1) to investigate whether different task formats would affect the performance and (2) to track developmental changes in this domain. Five to 16 year olds (N = 157) predicted the distances of a moving vehicle on the basis of its movement durations on both a horizontal and an inclined plane. The task formats involved: (1) nonverbal action tasks, (2) number-based missing-value word problems, and (3) verbal judgments. The majority of participants of all age groups based their reactions in the first two task types on the assumption of a linear relationship between time and distance—which is correct for motions with constant speed but incorrect for accelerated motions. However, in the verbal judgments that tapped conceptual understanding, children from the age of 8 years onwards correctly assumed that an object rolling down an inclined plane would accelerate. The role of the task format in evoking erroneous beliefs and strategies is discussed.  相似文献   

The predictive power of intuitive rules: A critical analysis of the impact of `more A–more B' and `same A–same B'     

Wim Van Dooren  Dirk De Bock  Dave Weyers  Lieven Verschaffel 《Educational Studies in Mathematics》2004,56(2-3):179-207

In the international community of mathematics and science educators the intuitive rules theory developed by the Israeli researchers Tirosh and Stavy receives much attention. According to this theory, students' responses to a variety of mathematical and scientific tasks can be explained in terms of their application of some common intuitive rules. Two major intuitive rules are manifested in comparison tasks: ‘More A—more B’ and ‘Same A—same B’. In this paper, we address two important questions for which the existing literature on intuitive rules does not provide a convincing research-based answer: (1) are the reasoning processes of students who respond in line with a given intuitive rule actually affected by that rule or by essentially other misconceptions (leading to the same answer), and (2) are individual students consistent in their choice of one of the intuitive rules when confronted with different, conceptually unrelated tasks? A test consisting of five comparison problems from different mathematical subdomains was administered collectively to 172 Flemish students from Grades 10 to 12. An analysis of students' written calculations and justifications suggested that the students were considerably less affected by the intuitive rules than their multiple-choice answers actually suggested. Instead, essentially different misconceptions and errors were found. With respect to the issue of individual consistency, we found that students who made many errors did not answer systematically in line with one of the two intuitive rules.  相似文献   

Children’s use of number line estimation strategies     

Dominique Peeters  Tine Degrande  Mirjam Ebersbach  Lieven Verschaffel  Koen Luwel 《European Journal of Psychology of Education - EJPE》2016,31(2):117-134

This study tested whether second graders use benchmark-based strategies when solving a number line estimation (NLE) task. Participants were assigned to one of three conditions based on the availability of benchmarks provided on the number line. In the bounded condition, number lines were only bounded at both sides by 0 and 200, while the midpoint condition included an additional benchmark at the midpoint and children in the quartile condition were provided with a benchmark at every quartile. First, the inclusion of a midpoint resulted in more accurate estimates around the middle of the number line in the midpoint condition compared to the bounded and, surprisingly, also the quartile condition. Furthermore, the two additional benchmarks in the quartile condition did not yield better estimations around the first and third quartile, because children frequently relied on an erroneous representation of these benchmarks, leading to systematic estimation errors. Second, verbal strategy reports revealed that children in the midpoint condition relied more frequently on the benchmark at the midpoint of the number line compared to the bounded condition, confirming the accuracy data. Finally, the frequency of use of benchmark-based strategies correlated positively with mathematics achievement and tended to correlate positively also with estimation accuracy. In sum, this study is one of the first to provide systematic evidence for children’s use of benchmark-based estimation strategies in NLE with natural numbers and its relationship with children’s NLE performance.  相似文献   

The development of students’ use of additive and proportional methods along primary and secondary school     

Ceneida Fernández  Salvador Llinares  Wim Van Dooren  Dirk De Bock  Lieven Verschaffel 《European Journal of Psychology of Education - EJPE》2012,27(3):421-438

This study investigates the development of proportional and additive methods along primary and secondary school. In particular, it simultaneously investigates the use of additive methods in proportional word problems and the use of proportional methods in additive word problems. We have also studied the role played by integer and non-integer relationships between the given numbers and the nature of quantities (discrete or continuous) in the development of these phenomena. A test consisting of additive and proportional missing-value word problems was solved by 755 primary and secondary school students (from fourth to tenth grade). The findings indicate that the use of additive methods in proportional situations increased during primary school and decreased during secondary school, whereas the use of proportional methods in additive situations increased along primary and secondary school. Moreover, the presence or absence of integer ratios strongly affects this behavior, but the nature of quantities only has a small influence on the use of proportional methods.  相似文献   

Students’ reported justifications for their representational choices in linear function problems: an interview study     

Ana Acevedo Nistal  Wim Van Dooren  Lieven Verschaffel 《Educational studies》2013,39(1):104-117

Thirty-six secondary school students aged 14–16 were interviewed while they chose between a table, a graph or a formula to solve three linear function problems. The justifications for their choices were classified as (1) task-related if they explicitly mentioned the to-be-solved problem, (2) subject-related if students mentioned their own characteristics as representational users, (3) context-related if contextual features surrounding the choice were mentioned and (4) representation-related if formal characteristics of the representations were pointed out. Justifications were mostly task- and subject-related, although contextual and representational features also played an important role. Some students (reportedly) tried to reconcile different (task-, subject-, context- and representation-related) factors before selecting a representation, which was interpreted as an attempt to use their meta-representational competence to make appropriate representational choices. The influence of the didactical contract and the experimental contract on students’ representational choices, as well as the tensions between them, are also discussed.  相似文献