Teaching Undergraduate Mathematics on the Internet

This is Part 2 of a two-part study of how APOS theory may be used to provide cognitive explanations of how students and mathematicians might think about the concept of infinity. We discuss infinite processes, describe how the mental mechanisms of interiorization and encapsulation can be used to conceive of an infinite process as a completed totality, explain the relationship between infinite processes and the objects that may result from them, and apply our analyses to certain mathematical issues related to infinity.     

现代数学基础理论中的一个不可解危机

从新构建的基础理论学的角度,讨论同时存在于现有集合论中两种性质上截然不同的“无穷集合”概念——潜无穷式的与实无穷式的无穷集合．指出现有数学中通常所使用的“无穷”概念极端的模糊不清,一方面混淆了原来的“有穷”概念,另一方面使人们无法理解与解析数学中与“无穷”相关的许多数量形式的性质,因此导致许多错误的数学行为．数学基础理论中这个致命缺陷是导致第二次与第三次数学危机的最根本原因．因此,只有对传统的“无穷”概念进行修正,构建新的“有穷-无穷”理论体系及相关的数量体系才是彻底解决第二次与第三次数学危机的惟一途径．     

统一无穷理论是消解悖论的典型案例

从数理辩证逻辑的观点看“统一无穷理论”,可以消解涉及无穷小的微积分悖论和涉及无穷大的集合论悖论,并能引出与有穷实数相互谐调一致的,涉及大小实无穷常数与最大最小潜无穷常数间的加、减、乘、除、乘方、开方的非标准数计算,由此讨论了“统一无穷理论”存在的合理性.     

Assessing high-school students’ ability to direct their learning

Theory on student learning provides that students are able to direct their learning when they have metacognitive knowledge about their own learning processes. In this article, a preliminary attempt to assess untrained high-school students’ metacognitive knowledge of learning processes as an ability through multiple-choice questions is reported. In three studies, item selection was established for ninth graders at the end of their school year. Also, in the final study the results showed that the ninth-grade students’ self-reported use of learning and studying strategies, study techniques, school learning and tiresome academic subjects related significantly to their metacognitive knowledge regarding learning processes. In the discussion, the practical consequences for school assessment are explicated and future research questions are raised.     

Some Historical Issues and Paradoxes Regarding the Concept of Infinity: An Apos-Based Analysis: Part 1

This paper applies APOS Theory to suggest a new explanation of how people might think about the concept of infinity. We propose cognitive explanations, and in some cases resolutions, of various dichotomies, paradoxes, and mathematical problems involving the concept of infinity. These explanations are expressed in terms of the mental mechanisms of interiorization and encapsulation. Our purpose for providing a cognitive perspective is that issues involving the infinite have been and continue to be a source of interest, of controversy, and of student difficulty. We provide a cognitive analysis of these issues as a contribution to the discussion. In this paper, Part 1, we focus on dichotomies and paradoxes and, in Part 2, we will discuss the notion of an infinite process and certain mathematical issues related to the concept of infinity.     

Examination involving students as peer examiners

The main interest in this article is students’ involvement in assessment as a part of growth towards self‐directedness in learning. In order to enhance students’ development of autonomy in learning, a project involving ‘older’ students as peer examiners for ‘younger’ students was designed and carried out. Students in the sixth semester in a PBL‐based Master’s program of Medical Biology participated, together with faculty, as examiners of fifth‐semester students. The examination and the assessment situation was carefully designed based on learning theories, empirical evidence and experiences underpinning student‐centred learning, especially in the form of PBL used at the faculty. The project was evaluated and analysed in order to understand students’ learning processes related to the responsibility for assessing peers. The situation of the peer examiners was interpreted based on their own experiences with statements from the students assessed and faculty involved in the assessment. Evaluations from six occasions, spring and fall, 2003–2005, were included in the study. The findings suggest that involving students in assessment as equal partners with faculty makes it is possible for students to apprehend the metacognitive competences needed to be responsible and autonomous in learning. The peer examiners experience motivation to learn about learning, they acquire tacit knowledge about assessment and they learn through being involved and trusted. The student‐centred educational context, which requires responsibility throughout the programme, is recognized as very important.     

Children’s Models about Colours in Nahuatl-Speaking Communities

This paper presents the development and structure of indigenous children’s ideas about mixing colours as well as their ideas about each colour, derived from their traditions. The children were interviewed both at school and outside it, and an educational proposal was implemented. Ideas expressed in the school context were analysed using the partial possible model, which states that the inferences and explanations used to describe a subject consist of constricting ideas, rules of correspondence, and a set of phenomenological inferences about processes. After identifying these components in the children’s ideas, we developed models to describe their conceptions about mixing colours. We employed a different approach to analyse children’s ideas related to their cultural context. The results showed that children change from a conception that focuses on colours as entities that do not change and as properties of objects (model 1) to the idea that colour represents a quality of substances or objects that can be modified by mixing colours (model 2). Cultural context analysis showed that stories are independent from one another and that they are not connected to colour mixing processes, only to the actions of colour on people. We concluded that students generate independent constructions between school and cultural knowledge.     

The Microcomputer and The Curriculum: A Critique

The classroom experience contains an infinite number of variables that cannot realistically be related to in any manageable teacher's manual. When manuals aim at being ‘practical’, what is produced is often something that looks like practicality, but is not. Curriculum‐writing needs a new approach, intended to educate teacher rather than students. Such curriculum‐writing can be described as ‘rehearsal curriculum’. A rehearsal curriculum allows the teacher to work through a process of learning, as a ‘rehearsal’ for directing his or her students through that same process. A rehearsal curriculum is written in a way that also motivates the teacher to learn.     

An Exploration of Secondary Students’ Mental States When Learning About Acids and Bases

This study explored factors of students’ mental states, including emotion, intention, internal mental representation, and external mental representation, which can affect their learning performance. In evaluating students’ mental states during the science learning process and the relationship between mental states and learning achievement, valid, reliable, and scalable measures of students’ mental states and learning achievement are needed. This paper presents the development of the Mental State Conceptual Learning Inventory (MSCLI) to identify students’ mental states before and after learning about acids and bases. This instrument is time efficient and convenient and can be administered to large student samples so that teachers and researchers can gain profound insights into their students’ learning of acids and bases in science class. The results of this study indicate that students’ mental states are highly correlated with their achievement. As a whole, low-achieving students tended to have negative emotions and low intentions, were not good at internal visualization, and were unable to interpret graphics and draw pictures. In contrast, high-achieving students had positive emotions and intentions when learning life-related topics about acids and bases, and were good at internal visualization and drawing and interpreting graphics.     

关于无穷多个无穷小量进行运算的探讨

文章对无穷多个无穷小的代数和及无穷多个无穷小的乘积进行了探讨,并搜集了一些现有教材和参考书中很少见到的独创的新例子,加深学生对无穷小性质的理解,进一步掌握教材中所叙述的有限个无穷小的代数和及乘积的结论。     

Why Inquiry? Primary Teachers’ Objectives in Choosing Inquiry- and Context-Based Instructional Strategies to Stimulate Students’ Science Learning

Studies have shown that there is a need for pedagogical content knowledge among science teachers. This study investigates two primary teachers and their objectives in choosing inquiry- and context-based instructional strategies as well as the relation between the choice of instructional strategies and the teachers’ knowledge about of students’ understanding and intended learning outcomes. Content representations created by the teachers and students’ experiences of the enacted teaching served as foundations for the teachers’ reflections during interviews. Data from the interviews were analyzed in terms of the intended, enacted, and experienced purposes of the teaching and, finally, as the relation between intended, enacted, and experienced purposes. Students’ experiences of the teaching were captured through a questionnaire, which was analyzed inductively, using content analysis. The results show that the teachers’ intended teaching objectives were that students would learn about water. During the enacted teaching, it seemed as if the inquiry process was in focus and this was also how many of the students experienced the objectives of the activities. There was a gap between the intended and experienced objectives. Hardly any relation was found between the teachers’ choice of instructional strategies and their knowledge about students’ understanding, with the exception that the teacher who also added drama wanted to support her students’ understanding of the states of water.     

Characteristics of an innovative learning environment according to students’ perceptions: actual versus preferred

An innovative learning environment is the current outcome of the constructivist approach, the essence of which is co-construction of knowledge in an Information and Communication Technology (ICT) environment. We examined how Israeli students perceived 10 characteristics of their classroom learning environment—student cohesiveness, teacher support, involvement, task orientation, investigation, cooperation, equity, differentiation, computer usage and young adult ethos. Particular foci were students’ perceptions of the actual state of their learning environment compared with the preferred state, and which characteristics predicted students’ cooperation. Participants were 1022 students in 33 classes from 12 computerised elementary and middle schools in Israel. Data were collected using the Technology-Rich Outcomes-Focused Learning Environment Inventory (TROFLEI). Results indicated a gap between the actual and the preferred states for all characteristics, although the scope of these gaps differed between elementary-schools students and middle-school students for certain characteristics. Structural equation modelling (SEM) analysis indicated that nine characteristics of the innovative environment in both actual and preferred states were related to cooperation, with these relations being primarily direct with the exception of teacher support and differentiation which had an indirect influence. Teacher support was mediated through student cohesiveness, involvement, equity and young adult ethos, whereas differentiation was mediated through investigation. Evaluation of the innovative learning environment might lead to better insights regarding the behaviours and needs of twenty-first century students in Israel’s education system. These insights could advance constructivist processes and teaching methods and bring the students to effective cooperative learning in an innovative learning environment.     

教育与自主创造幸福

教育的起点是人的生命。生命是有限与无限的统一、独特性与自主性的合一、创造性与超越性的契合。学校教育应遵循生命发展的规律，促进生命成长，引导生命走向自主与自控、和谐与无限，让生命在创造中找到永恒的幸福。具体而言，要培养学生积极情感；增强学生自我效能感；塑造学生健全人格；培养学生创造幸福的能力；提高教师自我创造幸福的能力。     

The Role of Intuitive Rules in Science and Mathematics Education

During the last two decades many researchers in mathematics and science education have studied students’ conceptions and ways of reasoning in mathematics and science. Most of this research is content‐specific. It was found that students hold alternative ideas that are not always compatible with those accepted in science. It was suggested that in the process of learning science or mathematics, students should restructure their specific conceptions to make them conform to currently accepted scientific ideas. In our work in mathematics and science education it became apparent that some of the alternative conceptions in science and mathematics are based on the same intuitive rules. We have so far identified two such rules: “More of A, more of B”, and “Subdivision processes can always be repeated”. The first rule is reflected in subjects’ responses to many tasks, including all classical Piagetian conservation tasks (conservation of number, area, weight, volume, matter, etc.) in all tasks related to intensive quantities (density, temperature, concentration, etc.) and in all tasks related to infinite quantities. The second rule is observed in students’, preservice and inservice teachers’ responses to tasks related to successive division of material and geometrical objects and in seriation tasks. In this paper, we describe and discuss these rules and their relevance to science and mathematics education.     

The longitudinal effects of teacher judgement and different teacher treatment on students’ academic outcomes

The self-fulfilling prophecy model of Brophy and Good was applied to the area of teacher judgement in order to disclose the processes of how teacher judgement of student achievement influences students’ future academic outcomes. It was assumed that achievement and achievement motivation might be affected through the mediating processes of student-perceived positive and negative teacher treatment, which represents students’ perceptions of teachers’ actions towards them. A sample of 294 Chinese fifth-grade students and their eleven English language teachers were tracked over one school year. Structural equation modelling showed that after controlling for prior achievement and achievement motivation, teacher judgement was directly related to students’ future achievement, expectancy for success and level of aspiration. These relations were partially mediated by perceived negative teacher treatment. Teacher judgement was related to students’ academic self-concept and pride in English learning, and fully mediated by both perceived positive and negative teacher treatment. The study supported the model of Brophy and Good and identified the specific processes of how teacher judgements can become self-fulfilled. Positive and negative teacher treatment had different mediating effects. Compared with positive teacher treatment, perceived negative treatment was related to student academic outcomes more extensively.     

Emotions in group work: insights from an appraisal-oriented perspective

Small group work is common practice in higher education. However, empirical research on students’ emotions related to group work is still relatively scarce. Particularly, little is known about students’ appraisals of a group task as antecedents of emotions arising in the context of group work. This paper provides a first attempt to systematically disentangle the relationship between individual differences in group work appraisals and emotions that arise in the context of a real-life group assignment. Beginning and end questionnaires (with an interval of 6 weeks) from 446 teacher education students who completed a mandatory, assessed group assignment within a university course were used to examine students’ multi-dimensional group work appraisals and emotional experiences. Findings showed that individual differences in emotions emerge as a function of appraisal processes and provided support for systematically treating students’ group work experiences as multi-dimensional, as appraisal dimensions have distinct implications for students’ emotional experiences.     

Analyzing the temporal evolution of students’ behaviors in open-ended learning environments

Metacognition and self-regulation are important for developing effective learning in the classroom and beyond, but novice learners often lack effective metacognitive and self-regulatory skills. However, researchers have demonstrated that metacognitive processes can be developed through practice and appropriate scaffolding. Betty’s Brain, an open-ended computer-based learning environment, helps students practice their cognitive skills and develop related metacognitive strategies as they learn science topics. In this paper, we analyze students’ activity sequences in a study that compared different categories of adaptive scaffolding in Betty’s Brain. The analysis techniques for measuring students’ cognitive and metacognitive processes extend our previous work on using sequence mining methods to discover students’ frequently-used behavior patterns by (i) developing a systematic approach for interpreting derived behavior patterns using a cognitive/metacognitive task model and (ii) analyzing the evolution of students’ frequent behavior patterns over time. Our results show that it is possible to identify students’ learning behaviors and analyze their evolution as they work in the Betty’s Brain environment. Further, the results illustrate that changes in student behavior were generally consistent with the scaffolding provided, suggesting that these metacognitive strategies can be taught to middle school students in computer-based learning environments.     

Effect of emotion-aware interventions on students’ behavioral and emotional states

The goal of this research is to investigate the effect of emotion-aware interventions on students’ behavioral and emotional states. To this end, we collected data from 12 students in the 9th grade in a high school in Turkey. The data collection took place in two sessions of an English Course. While the students were reading articles and solving relevant questions, our data collection application running in the background recorded the videos of the individual students through a camera and captured students’ screens in a non-intrusive manner. In total, we had 12.5 h of student data. We employed the human expert labeling process (HELP) (Aslan et al. in Workshop proceedings at international conference on intelligent tutoring systems (ITS), pp 156–165, 2016) to have the data labeled (150 h of data labeling in total). The data collection application was designed in a way that it also collected emotional self-labels (i.e., emotional states as self-reported by students at any time of learning). We leveraged emotional self-label information to suggest various real-time interventions for the students. The results obtained using the final expert labels showed that the percentage of the students’ Satisfied state was significantly higher after interventions. The results also demonstrated that although the interventions were triggered by the emotional states as self-labeled by the students and tailored to improve such states, there was a major positive impact of these interventions on students’ behavioral states. This preliminary study showed that even with a limited set of emotion-aware interventions based on self-labels, students’ states could be impacted positively. Conducting large-scale pilots leveraging more advanced interventions is a future direction for our research.     

Modification of a School Programme in the Deutsches Museum to Enhance Students’ Attitudes and Understanding

The study examines the nature, conditions, and outcomes of student learning from an organised guided tour in the Deutsches Museum in Munich. The instructional methods that best support students’ cognitive and affective learning as well as how students’ motivational and emotional states influence their achievement were investigated. A sample of 96 secondary school students took part in two different versions of a guided tour on an energy topic. The tours varied in the degree of support of students’ active involvement, group work, and the variety of general activities offered during the tour. The data collected indicate that both tour versions led to an increase in student understanding of the visit topic to nearly the same extent. However, the version stimulating students’ active participation, group work, and including a larger variety of activities aroused more positive attitudes. Students of the modified school programme showed higher interest and intrinsic motivation, felt more competent, and were less bored after the guided tour. In addition, the results suggest that students’ visit‐related emotional states predict the degree of their post‐visit topic understanding, even when demographics and prior knowledge are taken into consideration.     

Subdivision processes in mathematics and science

In the course of a research project now in progress, three successive division problems were presented to students in Grades 7–12. The first problem concerned a geometrical line segment, while the other two dealt with material substances (copper wire and water). All three problems involved the same process: successive division. Two of the problems (line segment and copper wire) were also figurally similar. Our data indicate that the similarity in the process had a profound effect on students' responses. The effect of the similarity in process suggests that the repeated process of division has a coercive effect, imposing itself on students' responses and encouraging then to view successive division processes as finite or infinite regardless of the content of the problem. It is possible to trace out, step by step, a more or less parallel process of development for the ideas of points and continuity and those dealing with atoms and physical objects in the child's conception of the ideal world. The only difference between these two processes is that to the child's way of thinking physical points or atoms still possess surface and volume, whereas mathematical points tend to lose all extension (though during the stages of development which concerns us here, this remains only a tendency.) (Piaget & Inhelder, 1948, pp. 126). Our first naive impression of nature and matter is that of continuity. Be it a piece of matter or a volume of liquid we invariably conceive it as divisible into infinity, and even so small a part of it appears to us to possess the same properties as the whole. (Hilbert, 1925, pp. 162).