Transformation and contextualisation: conceptualising students’ conceptual understandings of threshold concepts in calculus

Research on student learning in higher education suggests that threshold concepts within various disciplines have the capacity to transform students’ understanding. The present study explores students’ understanding in relation to particular threshold concepts in mathematics—integral and limit—and tries to clarify in what sense developing an understanding of those threshold concepts involves a transformation of understanding in relation to ways of thinking in mathematics. Drawing on data collected in interviews with students taking a basic course on calculus the analysis offers an initial characterisation of students’ understandings as algorithmic. It then proceeds to construct a more fine-grained theoretical account for how these understandings develop in the course of the interview, suggesting that the transformative aspects of threshold concepts may be conceptualised in terms of shifts in students’ contextualisations allowing the development of conceptions at different levels of abstraction simultaneously interacting to shape students’ awareness of the ways of thinking and practising in the subject.     

Three dimensions of reflective thinking in solving design problems: a conceptual model

Design tasks are omnipresent in our everyday lives. Previous research shows that reflective thinking is one of the critical factors in solving design problems. Related research has attempted to capture designers’ reflective thinking process. Yet a close inspection of designers’ reflective thinking taking place during their design process demands further effort. To understand designer’s reflective practice and to find better ways to promote novices’ reflective thinking in solving real-world design problems, a comprehensive model was developed. This model identified three dimensions to guide the understanding of designers’ reflective thinking during a design process: (1) the timing of reflection, indicating the points in the process where reflective thinking occurs, (2) the objects of reflection, showing the different types of objects that designers may reflect upon, and (3) the levels of reflection, referring to the different levels of designers’ reflection. This model provides for meaningful aspects of reflective thinking to be situated in a design process, which can guide educators and instructional designers in developing appropriate learning environments for facilitating novice and practicing designers’ reflective thinking. Moreover, the model can serve as a stepping stone for further research.     

Representational Issues in Students Learning About Evaporation

This study draws on recent research on the central role of representation in learning. While there has been considerable research on students’ understanding of evaporation, the representational issues entailed in this understanding have not been investigated in depth. The study explored students’ engagement with evaporation phenomena through various representational modes. The study indicates how a focus on representation can provide fresh insights into the conceptual task involved in learning science through an investigation of students’ responses to a structured classroom sequence and subsequent interviews over a year. A case study of one child’s learning demonstrates the way conceptual advances are integrally connected with the development of representational modes. The findings suggest that teacher-mediated negotiation of representational issues as students construct different modal accounts can support enriched learning by enabling both (a) richer conceptual understanding by students, and (b) enhanced teacher insights into students’ thinking.     

‘Thinking Schools, Learning Nations’ Implementation of Curriculum Review in Singapore

‘Thinking schools’ will be sites of learning for everyone declared the Singapore Prime Minister, Goh Chok and Minister of Education Teo Chee Hean’s in 1997 also spoke on the model of’ ‘thinking schools, learning nation’. Gardner’s model was used for the thinking school model in Singapore, in order to develop critical and creative thinking in students. This was to be done with the use of instructional technology as an enabling tool using a diversity of approaches including integrated project work. This paper reports on how one school went about changing approaches to teaching and learning by implementing integrated project work as a way of integrating the content areas of the curriculum, mathematics and science through English language, supported by the tools of instructional technology.     

Understanding the Dialectical Relations Between Everyday Concepts and Scientific Concepts Within Play-Based Programs

In recent times there has been an enormous interest in Vygotsky’s writing on conceptual development, particularly his insights on the differences between everyday and scientific thinking. In drawing upon cultural–historical theory, this paper seeks to examine the relations between everyday concepts and scientific concepts within playful contexts, such as preschools, with a view to better understanding how very young children develop conceptual understandings in science. This paper presents an overview of a study which sought to map the transformation and appropriation of scientific concepts within two early childhood settings. Approximately ten weeks of data gathering took place, with video recordings, field notes, photographic documentation, and child and teacher interviews for recording child concept formation within these naturalistic settings. The findings indicate that when teacher programs are more oriented towards concepts rather than materials, children’s play is focused on conceptual connections. Importantly, the study showed that: It was possible to map the multiple and dynamic levels or stratas of thinking that a child or group of children may exhibit within play-based contexts; An analysis of ‘unorganised heaps’ and ‘complexive thinking’ evident in conceptually or materially oriented play-based programs can be determined; the dialectical relations between everyday concepts and scientific concepts in play-based programs can be understood; and greater understanding about the nature of concept formation in situated playful contexts have been possible.     

The effect of two different cooperative approaches on students’ learning and practices within the context of a WebQuest science investigation

The goal of this study was to investigate the effect of two different cooperative learning approaches, namely, the Jigsaw Cooperative Approach (JCA) and the Traditional Cooperative Approach (TCA), on students’ learning and practices/actions within the context of a WebQuest science investigation. Another goal of this study was to identify possible problems that students face within the context of a WebQuest when following either approach and to provide suggestions for developing web-based learning tools that enable students to overcome these problems. The sample of the study consisted of 38 seventh-graders, who, according to their science teachers, had prior experience with TCA and JCA. All participants studied about the ecology, architecture, energy and insulation of CO₂-friendly houses through the use of a WebQuest science investigation. The data collection involved conceptual tests, screen–video captured data and interviews. Results revealed no differences between the two approaches, in terms of enhancing students’ understanding of concepts related to CO₂-friendly houses, because of (a) JCA students’ inability to apply one of the JCA components, namely, teaching one another about learning material they solely studied, and (b) the fact that the JCA students started applying the TCA after failing teaching one another in the context of JCA. Finally, a number of problems that students faced within the context of a WebQuest science investigation when following the JCA or TCA were identified.     

The relationship between students’ approaches to learning and the assessment of learning outcomes

The purpose of the present study is to gain more insight into the relationship between students’ approaches to learning and students’ quantitative learning outcomes, as a function of the different components of problem-solving that are measured within the assessment. Data were obtained from two sources: the revised two factor study process questionnaire (R-SPQ-2F) and students’ scores in their final multiple-choice exam. Using a model of cognitive components of problem-solving translated into specifications for assessment, the multiple-choice questions were divided into three categories. Three aspects of the knowledge structure that can be targeted by assessment of problem-solving were used as the distinguishing categories. These were: understanding of concepts; understanding of the principles that link concepts; and linking of concepts and principles to application conditions and procedures. The 133 second year law school students in our sample had slightly higher scores for the deep approach than for the surface approach to learning. Plotting students’ approaches to learning indicated that many students had low scores for both deep and surface approaches to learning. Correlational analysis showed no relationship between students’ approaches to learning and the components of problem-solving being measured within the multiple choice assessment. Several explanations are discussed.     

Towards a Critical Re-Appraisal of Ecology Education: Scheduling an Educational Intervention to Revisit the ‘Balance of Nature’ Metaphor

The ‘Balance of Nature’ metaphor is a pervasive idea in ecology. However, the scientific community acknowledged during the last decades that equilibrium conditions are rare, while disturbance events are not uncommon. We suggest that the exclusive teaching of the ‘Balance of Nature’ metaphor produces cultural, scientific and learning misconceptions about the structure and function of nature. We outline an exemplary educational intervention for high school students to exhibit that the use of computer simulations could serve important educational goals in ecology and environmental education, such as the liberation of the concept of ‘balance’ of its metaphysic burden, the comprehension of the dynamics and the systemic nature of ecological processes and the appreciation of the mutual relation between society and nature.     

Investigating strategies for using related cases to support design problem solving

Case-based learning has long been used to bring students into contact with the complexity of real-world situations. Despite this popularity and considerable history, research into how case analysis can support future problem-solving has been limited. The study reported in this paper investigated learners’ understanding of multimedia instructional design and development derived from the analysis of two richly detailed cases, and how this understanding then supported learners in their own design projects. A qualitative case study approach was used to follow a class of Masters students engaged in a technology-supported, case-based learning environment. Student work from case analysis, group project and reflective tasks was the key data source, complemented by interviews with students and their instructor, observations of class meetings, and the collection of online discussion list records and electronic resource files. The study found that the case analysis task raised learners’ awareness of design approaches and project management strategies, and that discussion and reflection play critical roles in developing students’ understanding. The study also highlighted some limitations of the case approach, suggesting the need for strategies that support learners’ thinking and reasoning.     

The learners’ experience of variation: following students’ threads of learning physics in computer simulation sessions

This article attempts to describe students’ process of learning physics using the notion of experiencing variation as the basic mechanism for learning, and thus explores what variation, with respect to a particular object of learning, that students experience in their process of constituting understanding. Theoretically, the analysis relies on analytic tools from the phenomenographic research tradition, and the recent group of studies colloquially known as the variation theory of learning, having the notion of experiencing variation as a key for learning at its core. Empirically, the study relies on video and audio recordings of seven pairs of students interacting in a computer-simulation learning environment featuring Bohr’s model of the atom. The data was analysed on a micro-level for the emergence of student-recognised variation, depicted in terms of ‘threads of learning’. This was done by linking variation around aspects of the object of learning present in the situation, and attended to by the students, to new ways of seeing—characterised as an expanding anatomy of awareness, and hence as learning. The students’ threads of learning are characterised in terms of two stages of learning progress: (1) discerning variation, and (2) constituting meaning from this experience of variation (experienced as holistically relevant in the students’ conceptual domain of physics and the Bohr model). Two groups of threads of learning were identified: one where the variation experienced by students was within an aspect of the object of learning, and one where variation was across several aspects.     

Components of Conceptual Ecologies

The theory of conceptual change is criticized because it focuses only on supposed underlying logical structures and rational process processes, and lacks attention to affective aspects as well as motivational constructs in students’ learning science. This is a vast underestimation of the complexity and diversity of one’s change of conceptions. The notion of conceptual ecology provides a context for understanding individuals’ conceptual change learning, as it is the environment through which all information is interpreted. This research investigated how high school students’ statements, made in answering questions, reflect selected components of their conceptual ecologies. Data for this study was collected from six interviews in which seven students took part. The data also include the science teacher’s profiles of each student, the students’ personal journals, their assignments, and their examinations and answers in class. The analysis presented will here include only those components that were represented in the discourse of the seven high school students who were interviewed. When students were asked questions, there was evidence of the engagement of the various components of conceptual ecologies. These components include: epistemological commitments, metaphysical beliefs, the affective domain and emotional aspects, the nature of knowledge, the nature of learning, the nature of conceptions, and past experience. Evidence from this study suggests that these components might function as constraints to learning. This study contributes to the field by expanding our knowledge of the components of high school students’ conceptual ecologies through its definition of the categories and themes associated with those components. In examining across the range of components, the study illustrates the variety and sources of science conceptions within high school students’ conceptual ecologies.     

A comparison of students’ reflective thinking across different years in a problem-based learning environment

Problem-based learning (PBL) is a constructivist approach to learning which is believed to promote reflective thinking in students. This study investigated how students in one particular institution developed in their reflective thinking habits—Habitual Action, Understanding, Reflection, and Critical Reflection—as they went through the daily practice of PBL. A 16-item questionnaire measuring the four levels of reflective thinking habits was administered to four cohorts of students: an incoming cohort, first-years, second-years, and third-years. First-year students rated themselves higher on Reflection and Critical Reflection, while third-years reported the highest levels of Habitual Action. Discriminatory and scatterplot analysis on the third year cohort revealed that while a proportion of students (47%) reported higher levels of Habitual Action with lower levels of Reflection, there was a small subgroup who also reported higher levels of both Habitual Action and Reflection. Overall, the results showed that PBL does promote the development of reflective thinking, particularly for first-year students, but that this development is not sustained consistently after that. This pointed to other possible factors that could hinder students’ development of reflective thinking in PBL.     

The nature of student predictions and learning opportunities in middle school algebra

In this article, we describe how using prediction during instruction can create learning opportunities to enhance the understanding and doing of mathematics. In doing so, we characterize the nature of the predictions students made and the levels of sophistication in students’ reasoning within a middle school algebra context. In this study, when linear and exponential functions were taught, prediction questions were posed at the launch of the lessons to reflect the mathematical ideas of each lesson. Students responded in writing along with supportive reasoning individually and then discussed their predictions and rationale. A total of 395 prediction responses were coded using a dual system: sophistication of reasoning, and the mechanism students appeared to utilize to formulate their prediction response. The results indicate that using prediction provoked students to connect among mathematical ideas that they learned. It was apparent that students also visualized mathematical ideas in the problem or the possible results of the problem. These results suggest that using prediction in fact provides learning opportunities for students to engage in mathematical sense making and reasoning, which promotes students’ understanding of the mathematics that they learn.     

Metacognition in chemical education: question posing in the case-based computerized learning environment

Posing questions about an article might improve one’s knowledge—a cognitive function, or monitor one’s thought processes—a metacognitive function. This study focuses on guided question posing while using a metacognitive strategy by 12th grade honors chemistry students. We investigated the ways by which the metacognitive strategy affected students’ skills to pose complex questions and to analyze them according to a specially designed taxonomy. Our learning unit, Case-based computerized laboratories, emphasizes learning through chemical case studies, accompanied by tasks, that call for posing questions to which the answer cannot be found in the text. Teachers equipped their students with a metacognitive strategy for assessing the quality of their own questions and characterizing them according to a three-component taxonomy: content, thinking level, and chemistry understanding levels. The participants were 793 experimental and 138 comparison chemistry students. Research instruments included interviews and case-based-questionnaires. Interviews with students revealed that using the metacognitive strategy the students had been taught, they were capable of analyzing the questions they generated with the taxonomy. The questionnaires showed that students significantly improved their question posing skill, as well as the complexity level of the questions they posed. A significant difference was found in favor of the experimental group students. Stimulating students to generate complex questions with a metacognitive strategy in mind enabled them to be aware of their own cognitive process and to self-regulate it with respect to the learning task.     

Characteristics and Levels of Sophistication: An Analysis of Chemistry Students’ Ability to Think with Mental Models

This study employed a case-study approach to reveal how an ability to think with mental models contributes to differences in students’ understanding of molecular geometry and polarity. We were interested in characterizing features and levels of sophistication regarding first-year university chemistry learners’ mental modeling behaviors while the learners were solving problems associated with spatial information. To serve this purpose, we conducted case studies on nine students who were sampled from high-scoring, moderate-scoring, and low-scoring students. Our findings point to five characteristics of mental modeling ability that distinguish students in the high-, moderate-, and low-ability groups from one another. Although the levels of mental modeling abilities have been described in categories (high, moderate, and low), they can be thought of as a continuum with the low-ability group reflecting students who have very limited ability to generate and use mental models whereas students in the high-ability group not only construct and use mental models as a thinking tool, but also analyze the problems to be solved, evaluate their mental models, and oversee entire mental modeling processes. Cross-case comparisons for students with different levels of mental modeling ability indicate that experiences of generating and manipulating a mental model based on imposed propositions are crucial for a learner’s efforts to incorporate content knowledge with visual-spatial thinking skills. This paper summarizes potential factors that undermine learners’ comprehension of molecular geometry and polarity and that influence mastery of this mental modeling ability.     

The impact of system specifics on systems thinking

Present and future social and ecological challenges are complex both to understand and to attempt to solve. To comprehend the complex systems underlying these issues, students need systems thinking skills. However, in science education, a uniform delineation of systems thinking across contexts has yet to be established. While there seems to be consensus on a number of key skills from a theoretical perspective, it remains uncertain whether it is possible to distinguish levels of systems thinking, and if so, how they would be determined. In this study, we investigated the impact of the specifics of a system on the skills and levels of systems thinking. We administered a 36-item multiple-choice test to 196 Grade 5 and 6 students. For our analysis, we followed a quantitative approach, applying a systems thinking model that incorporates the latest insights on the levels and skills of systems thinking in geography to the context of ecology. By following an Item Response Theory approach, we confirmed a set of systems thinking skills that are necessary to understand complex systems in ecology: identifying system organization, analyzing system behavior, and system modeling. We examined whether individual skill levels can be distinguished to determine whether a system's general principle or system-specific features cause difficulty for students. Our results indicate that system specifics, such as type of relation within ecosystems (e.g., predator–prey), appear to determine the formation of levels. Students struggled most with the difference between basic, direct cause-and-effect relationships and indirect effects. Once they understood the relevance of indirect relationships in moderately complex systems, a further increase in complexity caused little additional difficulty. Accordingly, we suggest that systems thinking should be examined from a variety of perspectives. To promote interdisciplinary learning, a systems thinking model that defines key commonalities across fields while leaving space for system specifics is needed.     

Psychological knowledge for teaching critical thinking: the agency of epistemic activity,metacognitive regulative behaviour and (student-centred) learning

This study considers the case of a tutor whose students repeatedly evidenced significantly superior critical thinking in summative assessment. For the purpose of surfacing appropriate pedagogical action to promote critical thinking (Bassey, Case Study Research in Educational Settings, 1999), the singularity of one tutor’s reported pedagogical practice was explored through focus-group discussion. Qualitative analysis of the data, theoretically informed by phenomenography, suggested that the tutor’s reported practice, when compared with that of two peers, revealed clear pedagogical intentions to be necessary for teaching critical thinking; and that these intentions can be explained through the literatures on epistemic activity, metacognitive regulative behaviour and student-centred learning. It is argued that a synthesised understanding of the literature that explores the nature and purpose of critical thinking —as outlined in the first part of this paper—is a prerequisite for constructing domain-specific pedagogical intentions for developing learners’ critical thinking, and that it is this extensive psychologically informed knowledge base which attenuates the risk of educationally important aspects of learning being overlooked. (De Corte, Learning and Instruction 10:249–266, 2000).     

The influence of framing on transfer: initial evidence from a tutoring experiment

This paper investigates the idea that the framing of learning and transfer contexts can influence students’ propensity to transfer what they have learned. We predicted that transfer would be promoted by framing contexts in an expansive manner in which students are positioned as having the opportunity to contribute to larger conversations that extend across time, places, people, and topics. A one-on-one tutoring experiment was conducted to test this hypothesis by manipulating framing as either expansive or its opposite (bounded) within a complex instructional learning ecology. We investigated the degree to which high school biology students transferred knowledge from a learning session about the cardiovascular system to a transfer-of-learning session about the respiratory system depending on framing condition. Consistent with the framing hypothesis, students in the expansive condition were generally more likely to transfer facts, a conceptual principle, and a learning strategy from one system to another.