Teaching mathematical proofs that rely on ideas from physics

Modeling stalagmite growth by first principles of chemistry and physics of calcite precipitation

In this paper a new learning algorithm, Lever Training Machine (LTM), is presented for binary classification. LTM is a supervised learning algorithm and its main idea is inspired from a physics principle: Lever Principle. Figuratively, LTM involves rolling a hyper-plane around the convex hull of the target training set, and using the equilibrium position of the hyper-plane to define a decision surfaces. In theory, the optimal goal of LTM is to maximize the correct rejection rate. If the distribution of target set is convex, a set of such decision surfaces can be trained for exact discrimination without false alarm. Two mathematic experiments and the practical application of face detection confirm that LTM is an effective learning algorithm.

Despite widespread agreement that proof should be central to all students’ mathematical experiences, many students demonstrate poor ability with it. The curriculum can play an important role in enhancing students’ proof capabilities: teachers’ decisions about what to implement in their classrooms, and how to implement it, are mediated through the curriculum materials they use. Yet, little research has focused on how proof is promoted in mathematics curriculum materials and, more specifically, on the guidance that curriculum materials offer to teachers to enact the proof opportunities designed in the curriculum. This paper presents an analytic approach that can be used in the examination of the guidance curriculum materials offer to teachers to implement in their classrooms the proof opportunities designed in the curriculum. Also, it presents findings obtained from application of this approach to an analysis of a popular US reform-based mathematics curriculum. Implications for curriculum design and research are discussed.

This paper investigates mathematics trainee teachers’ conceptions of proof writing in algebra at Copperbelt College of Education in Zambia. The ability to construct proofs seems to be an important skill for all mathematicians and those who have interest in mathematics education. However, literature shows that proof is widely acknowledged to be one of the most difficult aspects in the mathematics curriculum. The sample comprised of 35 male and 15 female third year mathematics trainee teachers. Data was collected by means of a self-administered questionnaire, a written diagnostic test and oral interviews. The three instruments were used to provide multiple views to triangulate the data. The results show that students seem to understand proof as a verification and confirmation tool. Many of them applied empirical approaches in their written proofs. They did not have sufficient knowledge of the right set of accepted statements such as axioms and could not identify which statements required proofs in mathematics. These findings suggest that students’ conceptions of proof were underdeveloped hence affecting their ability to produce valid proofs. These findings will assist in guiding curricula designers and teachers of proof related knowledge. DOI: 10.5901/mjss.2014.v5n14p331

When dealing with some problems like those regarding the sense and the nature of a mathematical proof, Euclid's Elements appear as a text where it seems suitable to analyse the deductive structure as well as the role that figures play within each argument. The novelty in Euclid's text does not rise from the fact that new theorems or unsolved problems are treated, but from the way in which the argument aiming to prove each proposition is provided. No example of this way of arguing towards a mathematical pro position is known in earlier mathematics, in fact, we could say that through the introduction of this particular form of proposition named theorem the Euclidean way of arguing became a sort of model. Although it is possible to find different sorts of proofs throughout the Euclidean text, it is not difficult to recognise in all of them a formal chain where each statement is connected with the following one. With this in mind we do not want to claim that there exists a sort of "Euclidean canon" providing the model that every proof should follow, but certainly every axiom and common notion, as well as every already proved statement used, are clearly indicated in the chain of arguments. Two kinds of questions arise when we recognise the role of proofs in Euclid's Elements', the first kind of questions ask whether the proof of a mathematical proposition is a kind of justification for its "truth" as if when the proof is provided the truth of the proposition should be accepted without any doubt. The second kind of questions are related to the different elements that make up each proof; for example the role that figures play in a geometrical proof. We should remark, above all, that a mathematical proof is a part of mathematics itself and, in this sense, we consider that it is not possible to

Proving is an essential mathematical activity that should play an important role in the mathematics classroom. However, performing a proof is a most challenging task for students. The nature of this challenge is discussed from a theoretical perspective. Starting with thoughts on the role of proof in mathematics, the article continues with reflections on limiting factors for performing proofs and the learning of proof in the mathematics classroom. Since empirical research gives evidence that guided exploration and a cherishing atmosphere can be regarded as beneficial for learning processes, this topic will be addressed in a further section.

Many taxonomies of levels of automation have been presented in the literature; however, the discrete and ordinal nature of these methods may limit reliable prediction of operator performance. This study defined an “automation rate” to quantity the level of automation in systems. To calculate the automation rate it is necessary to: classify all functions in the automation system according to stages of information processing, calculate the automation rate for each stage, set weights for these automation rates, and finally obtain the overall automation rate for the system. The practicality and feasibility of this model are verified through a case study analysis. In addition, this paper proposes a new relationship between the automation rate and operator situation awareness response, based on existing empirical research findings. Through case analysis and mathematical proof, the rationality of the form is demonstrated. This work lays the foundation for subsequent operator performance optimization analysis.

The persistence of ecological systems in changing environments requires energy, materials, and information. Although the importance of information to ecological function has been widely recognized, the fundamental principles of ecological science as commonly expressed do not reflect this central role of information processing. We articulate five fundamental principles of ecology that integrate information with energy and material constraints across scales of organization in living systems. We show how these principles outline new theoretical and empirical research challenges, and offer one novel attempt to incorporate them in a theoretical model. To provide adequate background for the principles, we review major concepts and identify common themes and key differences in information theories spanning physics, biology and semiotics. We structured our review around a series of questions about the role information may play in ecological systems: (i) what is information? (ii) how is information related to uncertainty? (iii) what is information processing? (iv) does information processing link ecological systems across scales? We highlight two aspects of information that capture its dual roles: syntactic information defining the processes that encode, filter and process information stored in biological structure and semiotic information associated with structures and their context. We argue that the principles of information in living systems promote a unified approach to understanding living systems in terms of first principles of biology and physics, and promote much needed theoretical and empirical advances in ecological research to unify understanding across disciplines and scales.

The national secondary school mathematics curriculum of Turkey, promulgated in 2013, encourages teachers to integrate dynamic geometry software into their classroom practices. One of the reasons for this request is that this kind of software offer teachers the opportunity to design learning environments in which students can experience the processes that constitutes mathematical exploration and proof. However, in order to exploit the possible advantages of integrating such software, teachers need the necessary technical knowledge about the software to be used and concrete examples of how it can be utilized as well. This study aims to contribute to the latter need by analyzing the solution steps of an open-ended geometry problem and exemplifying the role of the software in the solution process.

Two competing activist narratives dominate public conversation around the morality of abortion, yet little empirical research examines how women talk about the morality of their own abortion experiences. Drawing on a qualitative analysis of 156 accounts from an abortion storytelling website, I find that traditional pro-choice moral arguments invoking privacy and bodily autonomy were apparent but not dominant in women’s accounts. Three other frames emerged as ways to confront the problem of abortion as a morally controversial action: “abortion as morally unremarkable,” “abortion as morally problematic, but justified,” and “abortion as morally desirable.” These frameworks varied in the degrees to which they aligned with politically acceptable abortion narratives. Throughout these frameworks were a number of overlapping themes, including motherhood, responsibility, and religion. Drawing on theories of morality work and moral accounts, I posit that subjects are able to hold a “tension of opposites” while still maintaining cohesive narratives and presenting positive moral identities. For example, many women assigned their fetus a moral status as a life or potential life, yet ultimately felt other factors outweighed the obligation to sustain that life. I argue that this tension is a significant feature of morality work that warrants more attention.

The explicative aim of this research is to investigate the optics of proof format preferences held by prospective mathematics teachers (PMTs), in an effort to assess the effect that such preferences may have on their understanding of geometric proofs. The researchers employed a qualitative case study approach, in which six PMTs participated in Focus Group Discussions (FGDs) on the three styles of proof: paragraph proof, two-column proof and flow-chart proof. The methodology employed also focused on the formatting models by exploring the participants’ suggestions regarding each format in order to determine their cognitive responses as well as their preferences. The study exposes different preferences among PMTs which are a result of their personal cognitive style and experience with proofs. Some of the PMTs preferred the two-column format due to its systematic and logical nature, other PMTs preferred the storytelling nature of the paragraph proof format sponsored also flow-chart representation. These preferences bring out the requirement for the development of pedagogical strategies that employ different proof formats in order to meet the needs of different types of learners for effective understanding. In summary, this research demonstrates the need for customizing proof instruction to different students’ needs which can hopefully enhance PMTs’ responses and interactions with the concept of mathematical proofs. The practical outcomes show that teachers should not be afraid to think in more flexible witnessing forms of proof in order to aid students learning while minimizing the cognitive load in line with the principles of constructivist learning. It would be worthwhile however to extend this research and examine what these preferences would mean for the future and how those would fit within the context of online education with the overall aim of creating a better mathematics curriculum that is more effective and has a wider reach.

The style of investigation exemplified by Goethe’s experiments with color is often undervalued, but has repeatedly proved its worth.

This article looks at how social and participatory media can be used to strengthen interdisciplinary literacy and connects the multimodality of social environments with Middle‐Years Mathematics curriculum and delivery. The article reports on part of an eighteen months action research study in an Australian public high school within the author's two Year 8 Mathematics classes. The article provides student samples from within these classes as it contributes evidence and analyses that may help to inform teachers from across subject areas to recognise and implement literacy practices, not as an isolated classroom activity, but across their disciplines. By taking a more ‘social’ approach to designing projects for Middle‐Years Mathematics, this study supports concepts of literacy as a social, cognitive and cultural practice.

The purposes of this study were (1) to explore the relationship between mathematics education and the rural context and (2) to determine the usefulness of a social theory of learning (Wenger, 1998) to account for students' participation in advanced high school mathematics courses. Qualitative methods were used to gather data in a small, rural high school in the Pacific Northwest. Data gathered included a student survey, interviews with students and mathematics teachers, observations in mathematics classes, and documents related to the curriculum, the school, and the community. The findings suggest that high school mathematics education is related to the rural context in two ways. First, the small size produces characteristics unique to rural high schools. Specifically, a small number of mathematics teachers has considerable ability to shape the curriculum, instruction, and students' views of mathematics. The mathematics course offerings reflect a college preparatory curriculum with few mathematics electives offered. Students' experiences are quite similar and their beliefs about the nature and usefulness of mathematics are fairly uniform. Second, mathematics education is related to the socioeconomic aspects of a rural community. Students' knowledge of mathematics that is used in adults' work and daily life comes from experiences with parents and community members. The usefulness of mathematics, and advanced high school mathematics specifically, may not be apparent for students living in communities without mathematics-intensive jobs. Rural high school students studying advanced mathematics are likely exceeding the formal mathematical education level of their parents. The findings also indicated that Wenger's notions of identity and three modes of belonging provided a useful framework for understanding whether or not students participated in advanced mathematics classes. Students' participation in advanced math classes was related to their engagement in the day-to-day practice of mathematics education in the classroom, the place their engagement fit within the broader experiences of life in their imagination, and the alignment of their energies.

A guideline to study the feasibility domain of multi-trophic and changing ecological communities