Conceptual change in mathematics: From rational to (un)real numbers

From an educational point of view, mathematics is supposed to have a completely hierarchical structure in which all new concepts logically follow from prior ones. In this article we try to show that there are also concepts in mathematics which are difficult to learn because of problematic continuity from prior knowledge to new concepts. We focus on the problems of conceptual change connected with the learning of calculus and the shift from rational to real numbers. We demonstrate the difficulty of this conceptual change with the help of historical and psychological evidence. In the empirical study 65 students of higher secondary school were tested after a 40 hour calculus course. In addition, 11 students participated in individual interview. According to the results the conceptual change from a discrete to a continuous idea of numbers seems to be difficult for students. None of the subjects had developed an adequate understanding of real numbers although they had learned to carry out algorithmic procedures belonging to calculus. We discuss how appropriate recent theoretical ideas on conceptual change are for explaining learning problems in this domain. Also some educational implications are presented.     

Exploring the Probabilities of ‘Who Wants to be a Millionaire?’

This article discusses three probabilistic scenarios based on the television game show ‘Who Wants to be a Millionaire?’. These situations provide motivational opportunities for high‐school students to explore the concepts of expected value, permutations and independent events.     

Context-based learning in the middle years: achieving resonance between the real-world field and environmental science concepts

ABSTRACT

There is growing interest in how to engage middle school students in science to improve their enthusiasm for science and to arrest the decline in uptake in the senior years. Also, there is interest in improving students’ application of science to real-life situations, a requirement for international tests. One approach that offers hope for improving students’ connections between concepts and context is the context-based approach. Context-based units that connect canonical science with the real-world of the student’s local community have been trialled in the senior years but are new in the middle years. Research in senior classes has shown that students who were taught through a context-based approach demonstrated fluid transitions between the context and concepts in written work and student-student conversations. In the current ethnographic study we built on our previous work and investigated how students make connections between the environmental science concepts and the context of the weekly visits to the local creek. Students were immersed in the real-world context by completing an 11-week environmental science unit that required assessment of the health of a creek. Two assertions emerged; firstly, student-student conversations at the creek afforded students the opportunity for interconnections between environmental science concepts and the context (defined as resonance); and secondly, students’ written reports about the health of the creek demonstrated resonance. Furthermore, group work encouraged students the agency to complete sets of tasks that privileged visually obvious environmental science concepts such as pollution, identification of plants/animals or turbidity/flow rate.     

Secondary Students’ Stable and Unstable Optics Conceptions Using Contextualized Questions

This study focuses on elucidating and explaining reasons for the stability of and interrelationships between students’ conceptions about Light Propagation and Visibility of Objects using contextualized questions across 3 years of secondary schooling from Years 7 to 9. In a large-scale quantitative study involving 1,233 Korean students and 1,149 Singaporean students, data were analyzed from responses to the Light Propagation Diagnostic Instrument consisting of four pairs of items, each of which evaluated the same concept in two different problem situations. Findings show that only about 10–45 % of students could apply their conceptions of basic optics in contextualized problem situations giving rise to both stable and unstable alternative conceptions. Students’ understanding of Light Propagation concepts compared with Visibility of Objects concepts was more stable in different problem situations. The concepts of Light Propagation and Visibility of Objects were only moderately correlated. School grade was not a strong predictive variable, but students’ school achievement correlated strongly with their conceptual understanding in optics. The teaching and learning approach and education systems in the two countries may have had some influence on students’ conceptual understanding.     

Effects of a writing-to-learn tutorial programme in a higher education setting: An interactive qualitative study

ABSTRACT

Many higher education students embark on a study of accounting under the misconception that accounting requires a high level of proficiency in manipulating data and being good with numbers, while believing that linguistic competence, especially as it relates to writing, is of less importance. This article reports on a study that examined 15 managerial accounting and financial management (MAFM) students’ experiences of an 18-week writing intensive tutorial programme, based on writing-to-learn principles. Interactive qualitative analysis (IQA) informed the research design and data analysis. Following IQA protocols, nine affinities (themes) were generated. An account of how students’ experiences of a specific affinity, namely, written tasks, influenced their understanding of MAFM concepts is presented. The findings suggest that introducing writing-to-learn assignments contributed positively to students’ learning and understanding of MAFM concepts. This finding has implications for higher education pedagogy especially as it relates to the teaching of the accounting discipline.     

A scientific approach in the assessment of the school-based part of the teacher education programme in Sweden

This article aims to highlight how a scientific and critical approach is used in assessment dialogues during the last period of a practical, school-based teacher education programme. The result is based on 13 assessment dialogues conducted in a course at a Swedish university, where one of the course objectives is to, ‘in a scientific way, analyse teaching situations based on learning theories’. The assessment dialogues were analysed drawing from Bernstein’s concepts of ‘classification’, ‘framing’, ‘horizontal knowledge’ and ‘vertical knowledge’. The result shows that only in a minority of the assessment dialogues are students expected to use theory as an analytical tool and to critically examine their teaching practice. The theory is used in a more instrumental way to legitimize what is considered the ‘right way’ to teach. One conclusion is that the critical tradition of academia is seldom observed despite being clearly stated in the learning goals. The link between general academic knowledge and more school-based contextual knowledge is often missing or not made visible by the students or the teacher educators, and normative content is still clearly prominent.     

Teacher training by alternating classroom work and work analysis: from the perspective of a social conception of meaning and action

This article explores teacher training by the alternation of classroom work and work analysis using an approach based on a social conception of meaning and action. The advantage of this approach is that it allows the development of professional activity in preservice teachers (PTs) to be assessed by tracking how the reflective tools acquired in training evolve in work and/or work analysis situations. The concepts of ‘meaning’ and ‘expectation’ are helpful in discussing the empirical data from a research program designed to evaluate the potential for PTs’ professional development offered by the alternating work/analysis programs of French University Institutes of Teacher Training. The main conclusions concern the need to organize PTs’ professional experiences within a training network so that PTs are not left on their own to face the diverse and complex situations of daily professional exercise. The effective procedures are those that allow PTs to construct concrete and circumstance‐based expectations that the professional rules they have learned are still operational. These expectations help PTs to efficiently calibrate and interpret the meaning of their work experience, while also allowing them to prepare to act differently.     

The Boundary Between Finite and Infinite States Through the Concept of Limits of Sequences

In this article, attempts were made to examine students’ thinking about the concepts of infinity and their ideas about transiting from finite to infinite states through the concept of limits of sequences. The participants included 78 senior high-school students ranging in age between 17 and 19 years old. The data were collected through a questionnaire and an interview with all of the subjects. The findings showed that the students’ understanding of infinity is related to finite situations and many students consider infinite processes as a generalized form of finite processes. In the present study, the most common mistakes committed by students were related to consideration of infinity as a number and application of known finite results to infinite states.     

Meaningful Understanding and Systems Thinking in Organic Chemistry: Validating Measurement and Exploring Relationships

The purpose of this study was dual: First, to develop and validate assessment schemes for assessing 11th grade students’ meaningful understanding of organic chemistry concepts, as well as their systems thinking skills in the domain. Second, to explore the relationship between the two constructs of interest based on students’ performance on the applied assessment framework. For this purpose, (a) various types of objective assessment questions were developed and evaluated for assessing meaningful understanding, (b) a specific type of systemic assessment questions (SAQs) was developed and evaluated for assessing systems thinking skills, and (c) the association between students’ responses on the applied assessment schemes was explored. The results indicated that properly designed objective questions can effectively capture aspects of students’ meaningful understanding. It was also found that the SAQs can elicit systems thinking skills in the context of a formalistic systems thinking theoretical approach. Moreover, a significant relationship was observed between students’ responses on the two assessment strategies. This research provides evidence that students’ systems thinking level within a science domain is significantly related to their meaningful understanding of relative science concepts.     

STUDENTS’ UNDERSTANDING OF EQUILIBRIUM AND STABILITY: THE CASE OF DYNAMIC SYSTEMS

Engineering students in control courses have been observed to lack an understanding of equilibrium and stability, both of which are crucial concepts in this discipline. The introduction of these concepts is generally based on the study of classical examples from Newtonian mechanics supplemented with a control system. Equilibrium and stability are approached in different ways at the various stages of a typical engineering syllabus: at the beginning, they are mostly dealt with a static point of view, for example in mechanics, and are subsequently handled through dynamic analysis in control courses. In general, there is a little clarification of the differences between these concepts or the ways in which they are linked. We believe that this leads to much confusion and incomprehension among engineering students. Several studies have shown that students encounter difficulties when presented with simple familiar or academic static equilibrium cases in mechanics. Our study investigates students’ conceptions and misconceptions about equilibrium and stability through a series of questions about several innovative non-static situations. It reveals that the understanding of these notions is shaken when the systems being studied are placed in inertial or non-inertial moving reference frames. The students in our study were particularly uncertain about the existence of unstable equilibrium positions and had difficulty in differentiating between the two concepts. The results suggest that students use a velocity-based approach to explain such situations. A poor grasp of the above fundamental concepts may result from previous learning experiences. More specifically, certain difficulties seem to be directly linked to a lack of understanding of these concepts, while others are related to misconceptions arising from everyday experiences and the inappropriate use of physical examples in primary school.     

Some reflections upon teaching Sociology at a distance

This article reviews an introductory sociology course taught at a distance. After an outline of the course structure, the major teaching strategies adopted are discussed with an emphasis upon those which seek to foster sociological understanding by requiring students to explore links between their own social experience and the sociological concepts, theories and methods to which they are being introduced. Results from an evaluation study of students’ responses to these teaching strategies are also discussed.     

‘The balancing act’ – Irish part-time undergraduate students in higher education

While the numbers of part-time students has increased in higher education in Ireland, little is known about these students or about how they balance their study and other commitments. Drawing on a larger study on Irish students’ experiences in higher education, this article attempts to address this gap in research and reports on Irish part-time undergraduate students’ socio-demographic and institutional characteristics. It further explores the balance between part-time students’ academic study and their life situations. The article proposes some recommendations with regard to policy concerning part-time students, particularly in the area of financial support available to part-time learners. It also calls for expanding opportunities for participation in part-time study in the light of changing demography and provision of resources to facilitate students’ participation in higher education. Although the findings of this article broadly agree with those of international studies, more research is needed into the experiences of part-time students in higher education.     

基于核心素养的细胞的衰老与凋亡教学设计

以"细胞的衰老与凋亡"为例,通过创设问题情境、设置认知矛盾、小组讨论、社会话题探讨、迁移应用等活动,以知识为载体,传递正确的生命观念、严密的逻辑思维、严谨的科学探究精神、良好的社会责任感,培养学生的核心素养。     

Comparison of Traditional Instruction on Reflection and Rotation in a Nepalese High School with an ICT-Rich,Student-Centered,Investigative Approach

A teacher-centered, examination-driven instructional approach emphasizing knowledge of facts and standard methods through drill-and-practice without use of Information and Communications Technology (ICT) is still dominant in Nepalese high schools. In this article, we present a classroom study in which the traditional instructional approach has been replaced by an ICT-rich, student-centered, investigative approach in the context of teaching and learning basic concepts of reflection and rotation. Here, ICT refers to dynamic geometry software. Through a pretest-posttest control and experimental group research design, we compared the effects of both approaches on students’ understanding. A test was designed and used for investigating students’ alternative conceptions of reflection and rotation. The results showed that the experimental group outperformed the control group and there were indications of a lasting effect. Qualitative analysis indicated that all distinctive aspects of the experimental approach had positive effects on the students’ performance and learning experience. This study can also be considered as an evidence-based example of how one can, with limited ICT facilities, still achieve improvements in teaching and learning at a public high school in a developing country.     

‘If they can say it they can write it’: Inclusive pedagogies for senior secondary physical education

Pedagogical practices are fundamental to teachers' work, and in the spaces of schooling impact significantly on students' success and achievement (Evans, J. 1986. Physical Education, Sport and Schooling: Studies in the Sociology of Physical Education. London: Falmer Press.). This is especially the case for students from disadvantaged backgrounds who are deeply reliant on schooling for their educational resources. This article explores the interrelationships between pedagogical practices, the physical education curriculum at the senior secondary level and learning by both students and a teacher in a school located in an area of socio-economic disadvantage. Action research investigating a pedagogical redesign of a unit of ‘Skill Acquisition’ is the specific focus. Of key interest are pedagogical practices that incorporated opportunities to learn ‘about’ Skill Acquisition ‘through’ and ‘in’ movement. These practices attempted to develop and apply scientific literacies specific to the human movement sciences, which are important for academic success in senior secondary physical education. Findings reveal high student engagement, increasing utilisation of scientific literacies and application of new learning to life-world situations. We argue that pedagogical practices that integrate learning ‘about’ ‘through’ and ‘in’ movement disrupt default modes of teaching theoretical concepts in physical education, which diminish opportunities for academic success amongst students from low-socio-economic backgrounds.     

In between self-knowledge and school demands: policy enacted in the Swedish middle-year classroom

In this article, we focus on the interaction in a Year 5 classroom where students fill in a ‘self-evaluation form’ as a preparation for a forthcoming discussion on progress aiming at the production of an Individual Developmental Plan. Drawing on the theoretical concepts of fabrications and performativity, we understand this as an enactment of policy where both teacher and students become actors and subjects. From using document analysis together with conversation analysis as a methodological approach, we show how the ‘self-evaluation’ in interaction becomes a successful exercise in fabrications as teacher and student negotiate conceptions of the ideal student in relation to self-knowledge and school demands. The article is an empirically grounded contribution to the understanding of how policies are interpreted and made into being by local actors in everyday practices, in this case teachers and students in schools.     

Developing the ability to recontextualise cellular respiration: an explorative study in recontextualising biological concepts

In many science education practices, students are expected to develop an understanding of scientific knowledge without being allowed a view of the practices and cultures that have developed and use this knowledge. Therefore, students should be allowed to develop scientific concepts in relation to the contexts in which those concepts are used. Since many concepts are used in a variety of contexts, students need to be able to recontextualise and transfer their understanding of a concept from one context to another. This study aims to develop a learning and teaching strategy for recontextualising cellular respiration. This article focuses on students’ ability to recontextualise cellular respiration. The strategy allowed students to develop their understanding of cellular respiration by exploring its use and meaning in different contexts. A pre- and post-test design was used to test students’ understanding of cellular respiration. The results indicate that while students did develop an acceptable understanding of cellular respiration, they still had difficulty with recontextualising the concept to other contexts. Possible explanations for this ack of understanding are students’ familiarity with the biological object of focus in a context, the manner in which this object is used in a context and students’ understanding of specific elements of cellular respiration during the lessons. Although students did develop an adequate understanding of the concept, they do need more opportunities to practice recontextualising the concept in different contexts. Further research should focus on improving the strategy presented here and developing strategies for other core concepts in science.     

The Effects of a Teaching Perspective of Guided Internality on Intellectually Disabled Students,Locus of Control Orientation

The locus of control construct (LOC) and its implications for students’ behaviour are discussed. The effects of a teaching perspective designed to modify 110 intellectually disabled students’ LOC orientation was evaluated by analysing the data from a pre‐test, post‐test, follow‐up design study using: (1) an established LOC measure of students’ control of reinforcements for academic intellectual situations (MIARQ); (2) a new LOC measure of students’ expectancies of control of reinforcement for adaptive behaviour situations (KABLOC); and (3) a measure of students’ behaviour displayed in the classroom (BRS). The results for the students aged from 12 to 14 years indicated that the teaching perspective was effective in the longer‐term in modifying students’ LOC expectancies and overt behaviours. The results have positive implications for the education of intellectually disabled students.     

Making students part of the dataset: a model for statistical enquiry in social issues

This article reviews a flexible model for engaging students in statistical enquiry. Data from a student survey are first used to stimulate interest and introduce statistical concepts, while easing the transition to working with larger ‘real world’ secondary data.