Students’ understanding of the structure of deductive proof

While proof is central to mathematics, difficulties in the teaching and learning of proof are well-recognised internationally. Within the research literature, a number of theoretical frameworks relating to the teaching of different aspects of proof and proving are evident. In our work, we are focusing on secondary school students learning the structure of deductive proofs and, in this paper, we propose a theoretical framework based on this aspect of proof education. In our framework, we capture students’ understanding of the structure of deductive proofs in terms of three levels of increasing sophistication: Pre-structural, Partial-structural, and Holistic-structural, with the Partial-structural level further divided into two sub-levels: Elemental and Relational. In this paper, we apply the framework to data from our classroom research in which secondary school students (aged 14) tackled a series of lessons that provided an introduction to proof problems involving congruent triangles. Using data from the transcribed lessons, we focus in particular on students who displayed the tendency to accept a proof that contained logical circularity. From the perspective of our framework, we illustrate what we argue are two independent aspects of Relational understanding of the Partial-structural level, those of universal instantiation and hypothetical syllogism, and contend that accepting logical circularity can be an indicator of lack of understanding of syllogism. These findings can inform how teaching approaches might be improved so that students develop a more secure understanding of deductive proofs and proving in geometry.     

Students’ adoption of course-specific approaches to learning in two parallel courses

Research on students’ adoption of course-specific approaches to learning in parallel courses is limited and inconsistent. This study investigated second-semester psychology students’ levels of deep, surface and strategic approaches in two courses running in parallel within a real-life university setting. The results showed significant differences in course-specific adoption regardless of approach. However, the strategic approach seemed more stable. Sub-groups of students exhibited substantial variation: more than half of the students adopted similar approaches in both courses while large minorities showed great variability. Students scoring low on the surface approach, high on the deep approach or high on the strategic approach to learning were flexible in adopting different parallel approaches. However, students scoring high on the surface approach, low on the deep approach or low on the strategic approach seemed stable across contexts. This suggests that even smaller variations in teaching and learning environments can influence students’ approaches to learning.     

Students’ understanding and skills on voltage and current measurements using hands-on laboratory and simulation software

Education and Information Technologies - Skills competency is important especially in vocational and technical education. Thus, the aim of this research was to develop the student conceptual...     

Students’ understanding of the general notion of a function of two variables

In this study we analyze students’ understanding of two-variable function; in particular we consider their understanding of domain, possible arbitrary nature of function assignment, uniqueness of function image, and range. We use APOS theory and semiotic representation theory as a theoretical framework to analyze data obtained from interviews with thirteen students who had taken a multivariable calculus course. Results show that few students were able to construct an object conception of function of two variables. Most students showed difficulties finding domains of functions, in particular, when they were restricted to a specific region in the xy plane. They also showed that they had not fully coordinated their R³, set, and function of one variable schemata. We conclude from the analysis that many of the interviewed students’ notion of function can be considered as pre-Bourbaki.     

Students’ images and their understanding of definitions of the limit of a sequence

There are many studies on the role of images in understanding the concept of limit. However, relatively few studies have been conducted on how students’ understanding of the rigorous definition of limit is influenced by the images of limit that the students have constructed through their previous learning. This study explored how calculus students’ images of the limit of a sequence influence their understanding of definitions of the limit of a sequence. In a series of task-based interviews, students evaluated the propriety of statements describing the convergence of sequences through a specially designed hands-on activity, called the ɛ–strip activity. This paper illustrates how these students’ understanding of definitions of the limit of a sequence was influenced by their images of limits as asymptotes, cluster points, or true limit points. The implications of this study for teaching and learning the concept of limit, as well as on research in mathematics education, are also discussed.     

Three concepts or one? Students’ understanding of basic limit concepts

In many mathematics curricula, the notion of limit is introduced three times: the limit of a sequence, the limit of a function at a point and the limit of a function at infinity. Despite the use of very similar symbols, few connections between these notions are made explicitly and few papers in the large literature on student understanding of limit connect them. This paper examines the nature of connections made by students exposed to this fragmented curriculum. The study adopted a phenomenographic approach and used card sorting and comparison tasks to expose students to symbols representing these different types of limit. The findings suggest that, while some students treat limit cases as separate, some can draw connections, but often do so in ways which are at odds with the formal mathematics. In particular, while there are occasional, implicit uses of neighbourhood notions, no student in the study appeared to possess a unifying organisational framework for all three basic uses of limit.     

Students’ Expectations of Teaching in Undergraduate Physics

This article reports on a phenomenographic‐based study of the expectations of teaching among undergraduate physics students. Data are drawn from a range of course‐contexts at each of two quite different universities — one South African and one Swedish — and five qualitatively different expectations of physics teaching are identified and exemplified. These range from presenting knowledge to facilitating personal development and agency. Implications of possible mismatches between students’ and lecturers’ expectations of physics teaching are discussed in relation to students’ learning experiences.     

Students’ Views of Learning in Vocational Education

The aim of this study was to investigate students’ conceptions of learning and approaches to learning within vocational education. A group of car‐mechanic students (n = 30) were interviewed at the end of their programme. The phenomenographic approach was used as the point of departure, and, as a result of the analysis, qualitative similarities and differences in students’ conceptions and approaches were found. The distribution of conceptions and approaches showed that half of students were classified as representing a quantitative view of learning and as favouring a surface approach. However, in order to encourage students to develop a qualitative view of learning and to use deep approaches, more attention must be paid to the ways in which students conceive learning within the school context and how they actually do in order to meet the demands they experience within their education.     

Cultivation of Students’Intercultural Awareness in CET

Language is the carrier of culture and it is also an integral part of culture. Therefore, learning English requires us not only master the English language knowledge and skills, but also understand t...     

Students’ Experiences of Meaningful Situations in School

This paper focuses on school situations students experienced as meaningful and how these experiences can guide educational improvement. Participants included 15 students in grade 3 from a Swedish school. In this qualitative study, the data consisted of drawings, multimodal productions, interviews, and field notes. The analysis resulted in four themes: Having the opportunity to learn in different spaces; Being free and able to participate; Experiencing caring and sharing, and Recognizing one’s own growth and achievement. The findings suggest that situations students find meaningful involve aspects of both learning and wellbeing. The practical implication for these results is that student-generated qualitative data can help indicate needs for educational improvement.     

On Development of Students’Listening in Class

Listening is considered to be one of the most difficult skills in English learning.And it is obvious that the learners do not have enough and effective listening practice in a scientific way.This thesis will focus on listening.The author mainly analyzes how to develop the listening strategies to improve the ability of listening comprehension.     

Students’ understanding of a groundwater system and attitudes towards groundwater use and conservation

The purpose of this study was to investigate students’ understanding of groundwater as a system, and to further examine whether and how their understanding is related to attitudes towards groundwater use and conservation. A total of 676 junior high school students completed a groundwater survey including a drawing activity and a questionnaire. An additional group of university students with environmental science majors were recruited for the survey for the purpose of determining reasonable expectations and validating interpretations of the results. Analysis of student drawings indicated that students generally held an incomplete and unconnected understanding of the groundwater system. Correlation analysis showed that student levels of understanding were associated with their perceptions of environmental impacts as well as environmental concern regarding groundwater problems. These findings re-emphasised the importance of promoting students’ understanding of environmental systems, as well as their ability to think systemically.     

On Stimulation of Students’Interests in English Learning

As the old saying goes that interest is the best teacher and it is quite true that the students’initiative in learning is diffi cult to spur without interest.English is the most widely used language in the world,and people are aware of the importance and necessity in English learning.But English is a second language to us,so learning it is more difficult than the acquiring of our na tive language.Interest is the internal power to spur students’learning.So how to spur the students’s interest in engliash learning is the main prblem discussing in it.     

Students ’ perceptions of science classes in the Philippines

This study used a modified version of the Perception of Science Classes Survey (Kardash & Wallace, 2001) to survey 7,885 grade school and high school students in different provinces in the Philippines regarding their perceptions of their science classes. Factor analysis revealed five dimensions of the students ’ perceptions that relate to different aspects of the teacher ’s pedagogy and the learning environment created by the teachers: (a) Learner-Centered Pedagogy, (b) Science Inquiry Activities, (c) Positive Affect and Attitudes, (d) Grades as Feedback, and (e) Support for Self-Learning and Effort. Factor scores were compared across grade levels and genders. The results indicate a decrease in science inquiry activities and the use of grades as feedback in the higher grades, but an increase in support for self-learning and effort, and also positive affects and attitudes. These trends were discussed in relation to possible problems related to teacher practices that may contribute to low student achievement levels in science. This research was part of a research project commissioned by the Japan International Cooperation Agency-Philippines.     

Variation in Students’ Experiences of the ‘Oxford Tutorial’

This paper examines Oxford University students’ conceptions of the role of the tutorial in their learning. An analysis of interviews with 28 students constituted four qualitatively different conceptions of the ‘Oxford Tutorial’. These ranged from the tutorial involving the tutor explaining to the student what the student did not know, to the tutorial involving the tutor and the student in exchanging different points of view and both coming to a new understanding of the topic under discussion. These different conceptions also appeared to be related to variations in students’ views of the role of the work done in preparation for the tutorial, their view of the student’s and tutor’s roles in the tutorial, and the conception of knowledge that students adopted in relation to the tutorial. The implications of this study are discussed in terms of the relations between students’ conceptions of tutorials and their anticipated learning outcomes and its implications for contexts outside of Oxford in terms of students’ conceptions of academic tasks.     

Students’ experience of synchronous learning in distributed environments

This article reports on a two-year ethnographic study of learners participating in multi-site, graduate-level education classes. Classes sometimes met face-to-face in the same physical location; at other times part of the class met physically elsewhere. Yet all were linked through the virtual space. Ethnographic analysis of four data types explored how the instructor and students were able to interact through videoconferencing technologies. Most of the interaction occurred between the local and distance learners by way of cultural guides, local students assigned to host a distance learner through Google Video chat. The distance learners were able to receive real-time attention from the instructor and were able to share differing perspectives that contributed to increased satisfaction in the course. These interactions allowed for a dynamic collaborative effort among a diverse set of actors in the field of education.