Problem solving and chemical equilibrium: Successful versus unsuccessful performance

The purpose of this study was to describe the problem-solving behaviors of experts and novices engaged in solving seven chemical equilibrium problems. Thirteen novices (five high-school students, five undergraduate majors, and three nonmajors) and ten experts (six doctoral students and four faculty members) were videotaped as they individually solved standard chemical equilibrium problems. The nature of the problems was such that they required more than mere recall or algorithmic learning and yet simple enough to provide the novices a reasonable chance of solving them. Extensive analysis of the think-aloud protocols produced 27 behavioral tendencies that can be used to describe and differentiate between successful and unsuccessful problem solvers. Successful solvers' perceptions of the problem were characterized by careful analysis and reasoning of the task, use of related principles and concepts to justify their answers, frequent checks of the consistency of answers and reasons, and better quality of procedural and strategic knowledge. Unsuccessful subjects had many knowledge gaps and misconceptions about the nature of chemical equilibrium. Even faculty experts were sometimes unable to correctly apply common chemical principles during the problem-solving process. Important theoretical concepts such as molar enthalpy, heat of reaction, free energy of formation, and free energy of reaction were rarely used by novices in explaining problems.     

基于PISA2021数学素养的数学推理与问题解决

PISA测试结果的每一次公布都会引起世界的瞩目,各国政府及相关教育政策决策者会依据其结果对其相关教育政策作出调整。在正式实施测试之前,OECD会提前公布相关测试框架,这会在一定程度上影响未来的教学与评价走向。PISA2021测试框架最为显著的一个变化体现在数学素养定义中的数学推理,侧重在数学推理的介绍及其与问题解决的关系。通过对PISA2021的分析发现,数学推理包括演绎推理和归纳推理,贯穿问题解决的全过程,所有数学活动的展开都围绕数学推理而进行。     

2021年PISA数学——来自CCR的分析

数学是理解世界、公民身份和经济增长的基石。为了满足全社会对教育的需求,21世纪教育应该注重对知识理解的深度和多样性的培养。PISA关于数学能力的测试中,最重视学生运用数学推理来解决问题的能力。我们建议扩展数学过程的描述(表述、应用、解释、评估),并在PISA数学框架内确定这些处理过程为数学建模的主要组成部分,其中有七个最常用于寻找正确推理方法的推理工具:比较、比例推理、应用乘法量表、拆分、归并、由简入繁、概率推理和逻辑推理。PISA数学的素养领域涉及形状与空间、变化与关系、不确定性与数据、数量等,还要特别注意创造性思维能力、品格和元认知技能的培养。     

evolving research of mathematics teacher educators: the case of non-standard issues in solving standard problems

This study describes the learning of researchers who engage in mathematics teacher education as an integral part of their practice. As teacher educators working with teachers on the subject of proportional reasoning, the authors reflected on teachers solutions to a standard problem and analyzed answers that would conventionally be considered incorrect. This exploration showed that some incorrect answers made sense, were based on problem situation analysis, and brought attention to the fact that conventional formal answers were given without much deliberation on their meaning in the situation. This insight prompted a second research phase in which teachers discussed and explained alternative solutions, and developed deeper analysis of problem situation in solutions that had been correct in the first place. The importance of making teachers aware of the nature of alternative solutions was further exhibited in a third research phase in which teachers evaluated childrens answers to the same problem. The pedagogical insight that emerged stressed the importance of making teachers aware of the tension between an almost automatic application of a mathematical model, and of analyzing problem situation during problem solving. In addition, the researchers developed better understanding of the mathematical challenge associated with the proportional reasoning problem, a stronger awareness of the role of sensitivity to their learners (the teachers), and of the role of reflection.     

Finnish pre-service teachers’ and upper secondary students’ understanding of division and reasoning strategies used

In this paper, we focus on Finnish pre-service elementary teachers’ (N?=?269) and upper secondary students’ (N?=?1,434) understanding of division. In the questionnaire, we used the following non-standard division problem: “We know that 498:6?=?83. How could you conclude from this relationship (without using long-division algorithm) what 491:6?=?? is?” This problem especially measures conceptual understanding, adaptive reasoning, and procedural fluency. Based on the results, we can conclude that division seems not to be fully understood: 45% of the pre-service teachers and 37% of upper secondary students were able to produce complete or mainly correct solutions. The reasoning strategies used by these two groups did not differ very much. We identified four main reasons for problems in understanding this task: (1) staying on the integer level, (2) an inability to handle the remainder, (3) difficulties in understanding the relationships between different operations, and (4) insufficient reasoning strategies. It seems that learners’ reasoning strategies in particular play a central role when teachers try to improve learners’ proficiency.     

University Students' Strategies for Constructing Hypothesis when Tackling Paper-and-Pencil Tasks in Physics

The study we present tries to explore how first year engineering students formulate hypotheses in order to construct their own problem solving structure when confronted with problems in physics. Under the constructivistic perspective of the teaching–learning process, the formulation of hypotheses plays a key role in contrasting the coherence of the students' ideas with the theoretical frame. The main research instrument used to identify students' reasoning is the written report by the student on how they have attempted four problem solving tasks in which they have been asked explicitly to formulate hypotheses. The protocols used in the assessment of the solutions consisted of a semi-quantitative study based on grids designed for the analysis of written answers. In this paper we have included two of the tasks used and the corresponding scheme for the categorisation of the answers. Details of the other two tasks are also outlined. According to our findings we would say that the majority of students judge a hypothesis to be plausible if it is congruent with their previous knowledge without rigorously checking it against the theoretical framework explained in class.     

重建对话教学的方法论

对话教学把教学方法理解为智慧或问题解决、社会互动与合作、个人风格与个性独特性三者的合金。重建对话教学的方法论意味着走出技术主义的窠臼,让教学建基于批判意识和反思性实践,把主题探究、问答、会话、辩论、对话性讲授、交往性沉默、个人方法等方法论要素创造性融为一体,化成以智慧和民主为核心的教学行动。     

All good readers are the same,but every low-skilled reader is different: an eye-tracking study using PISA data

PISA results show that a considerable number of 15-year-old pupils after 8 to 10 years of schooling have a low level of functional reading literacy, as defined in the PISA framework. While PISA results help identify the level of reading competency, they do not reveal what might be the reasons why some students fail to solve the tasks. One way to explore the difficulties students encounter while solving PISA reading tasks is to track their eye movements during reading. The main aim of this study was to explore the similarities and differences in eye movement patterns between students with high and low scores on PISA reading tasks. A sample of 92 students took part in the pre-test, which was based on PISA items, and administered to identify groups of students with high and low PISA reading scores. Based on student pre-test results, 20 students were selected for the main, eye-tracking test—10 participants with low scores and 10 with high scores. The eye-tracking test consisted of four different released PISA reading tasks, three of them continuous and one non-continuous. The continuous items were followed by one multiple-choice question each at L1, L2, and L3 levels of difficulty. The non-continuous text was followed by three multiple-choice questions (also L1–L3).Three main findings were the following: regarding saccadic amplitudes, the reading was found to be less fluent for the low-skilled group; according to the heat maps, they had difficulty finding the relevant material; and taking into account standard deviations of eye-tracking measures, the variability was found to be larger in this group. Taken together, the findings of this study show that the PISA results differentiating low- and high-scoring groups go hand in hand with insights from more fine-grained eye-tracking measurements.     

Zwischen Schule und Universität: Argumentation in der Mathematik

Several studies show that university students in Germany still have problems in reasoning mathematically although this already should be fostered at high school since the implementation of standards for school mathematics. Mathematical argumentation is a core competence and highly important, especially in academic mathematics. To foster mathematical argumentation at the beginning of university studies, competence models are needed which give more detailed insights in the skills that are necessary for reasoning. As mathematical argumentation is a complex process, especially at the higher secondary level or at university, many little steps are needed to complete a competence model for argumentation at the secondary–tertiary transition gradually. A possible step can be to initially identify several aspects of mathematical argumentation competence that influence the reasoning quality. The empirical basis for identifying those aspects is a cross-sectional study with 439 engineering students who participate in a transition course in mathematics. We address the following questions: (1) how is the quality of student’s reasoning? (2) Which kind of arguments do students use? (3) What resources do students who reasoned correctly use for solving the problems? (4) Does the content of the tasks play an important role? The results show a great influence of the content on the reasoning quality, especially if the content is abstract or concrete. Argumentation quality of students decreases with an increasing level of abstraction of the content. Furthermore, the results reveal that students often use routines for solving the problems. That indicates that procedural approaches still play an important role in school mathematics. If procedures could be used for solving the tasks, students are more successful. Competence models for mathematical argumentation at the beginning of the tertiary level should, therefore, include these factors.     

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     

Towards detection of learner misconceptions in a medical learning environment: a subgroup discovery approach

Ill-structured problems, by definition, have multiple paths to a solution and are multifaceted making automated assessment and feedback a difficult challenge. Diagnostic reasoning about medical cases meet the criteria of ill-structured problem solving since there are multiple solution paths. The goal of this study was to develop an adaptive feedback mechanism that is capable of identifying and responding to novice physician misconceptions by mining the log trace data of user interactions in BioWorld, a computer-based learning environment designed to support medical students in regulating their own diagnostic reasoning. We applied a series of text pre-processing algorithms to extract the linguistic features that characterized symptoms identified by 30 medical students solving three endocrinology cases: diabetes mellitus (type 1), Pheochromocytoma, and Hyperthyroidism. A subgroup discovery mining algorithm was applied in two steps. First, rules were induced to predict an incorrect diagnosis submitted as the final solution for a case on the basis of symptoms highlighted by medical students as being pertinent, that were in fact not pertinent. Second, rules were induced to predict a distractor hypothesis (an incorrect hypothesis listed as the most probable) during the differential diagnosis process while solving the case. The rule set discovered through the subgroup discovery task was shown to predict both incorrect and distractor hypotheses set by novice physicians while solving the case. We discuss the implications in terms of developing an adaptive feedback mechanism that can detect physicians’ misconceptions and errors during problem-solving as a means to deliver prompts and feedback that promote the acquisition of metacognitive monitoring and control skills.     

Conditional logic abilities on the four-card problem: Assessment of behavioral and reasoning performances

The present study assessed the behavioral and the reasoning performances of 507 school and university students on the four logical principles of logical detachment, particular conversion, particular inversion, and particular contraposition. An adapted version of Wason's four-card problem was administered to all students in paper-and-pencil format and in group settings. Students were asked to respond to the logical questions and to justify their answers. Nine logical reasoning categories were identified. The results of behavioral and reasoning performances were compared across grade levels. Results indicated that many students who responded behaviorally correctly to the logical questions, provided incorrect reasoning justifications. The percentages of students who considered a conditional statement as hypothetical increased with age. No sex differences on the behavioral and the reasoning performances were reported.     

Adolescents' collaboration in the classroom: Do peer relationships or gender matter?

Peer collaboration can be a useful tool in a school classroom to help students perform at their best. With whom should students be paired, though? Previous research yields inconsistent findings regarding whether the benefits of peer collaboration depend on the gender or friendship of collaborators. We paired students with a same‐gender friend or a nonfriend in their classroom to examine whether friendship and specific dimensions of relationship quality were important for understanding adolescent (N = 132 high‐school students) boys' and girls' performance on a scientific reasoning task. Dimensions of relationship quality were related to task performance with greater perceived conflict predicting poorer performance. Girls outperformed boys, but the difference was marginal and nonsignificant after accounting for dimensions of relationship quality. Friends' and nonfriends' performance was similar. Results are informative for educators who use collaboration as an instructional technique and for other professionals who work to support the development of effective reasoning and problem‐solving skills among adolescents. © 2008 Wiley Periodicals, Inc.     

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.     

Student ability in identifying reasons for different points of view: A learning activity conducted with a group of Japanese High School Students

Problem solving as a skill is emphasized within the revised Japanese Course of Study for schools. This experimental study, conducted among Japanese high school students, evaluates an effort to develop student ability to identify reasons fordifferent points of view, not just personally favored views. First, the ability of all students was tested using a reading literacy question from the 2000 PISA test, which measures the reading skills of evaluation and reflection. The experimental group then participated in a developmental learning activity. After the activity, all students were retested using the same PISA question. Nothing conclusive was found regarding the general effectiveness of the lesson activity, but two insights were gained. It was found that student self-awareness of the importance of the learning task and interactive learning are important to consider.     

Students' Understandings of Gravity in an Orbiting Space-Ship

We report on an investigation of students' ideas about gravity after a semester of instruction in physics at university. There are two aspects to the study which was concerned with students' answers to a carefully designed qualitative examination question on gravity. The first aspect is a classification of the answers and a comparative study of the ways the problem was tackled by two large groups of students who had different backgrounds in physics and were exposed to different teaching styles. The second aspect is to investigate how students link concepts to solve the problem. We used a phenomenographic analysis of student responses to extract patterns of reasoning and alternative conceptions behind the solutions. We found no differences between the classes of answers given by students in the two courses. Our analysis also identifies a hierarchy in the complexity of the hypothetical reasoning pathways, which we interpret as reflecting the ways in which students may link concepts and resolve conflicts as they solve the problem. The hypothetical reasoning pathways may help educators to develop instructional material or lecture room dialogue in order to tease out key issues. An unexpected finding is that there is a discrepancy between our conclusion that the two groups of answers are similar and the distribution of marks awarded by the examiner – which implies that the quality of the answers is different for the two groups.     

Leveraging cultural resources through teacher pedagogical reasoning: Elementary grade teachers analyze second language learners' science problem solving

Grounded in teacher professional development addressing the intersection of student diversity and content area instruction, this study examined school teachers' pedagogical reasoning complexity as they reflected on their second language learners' science problem solving abilities using both home and school contexts. Teachers responded to interview questions after watching a video of one of their students engaged in a science problem solving task. Over a 5-year period, 206 teacher interviews were conducted with a total of 133 teachers. Results indicated significant differences across the dimensions of pedagogical reasoning complexity as teachers expressed both deficit and resource oriented thinking.