• School Mathematics
• /
• v.14 no.3
• /
• pp.297-313
• /
• 2012

The purpose of this paper is to discuss the potential and to examine the effect of narrative, as an alternative approach to teach formal proof in more easier and comprehensible way. Identifying the key elements of narrative in proof, we constructed a narrative that derives the equation of function translation. We examined the effect of teaching through the narrative, in comparison with teaching the corresponding proof, on low-achieving students' instrumental understanding and relational understanding of function translation. Since we found no statistically significant differences between the experimental and the comparison group, this study could not conclude that teaching through the narrative was more effective than teaching the corresponding proof. But there were some qualitative differences in the relational understanding responses and the evaluation of the teaching between two groups. These findings suggested some potential of narratives that complement the formal proof.

• Proceedings of the Korean Nuclear Society Conference
• /
• 1997.05a
• /
• pp.153-158
• /
• 1997

This paper describes a formal safety analysis technique which is demonstrated by performing empirical formal safety analysis with the case study of beamline hutch door Interlock system that is developed by using PLC(Programmable Logic Controller) systems at the Pohang Accelerator Laboratory. In order to perform formal safety analysis, we have built the Z formal specifications representation from user requirement written in ambiguous natural language and target PLC ladder logic, respectively. We have also studied the effective method to express typical PLC timer component by using specific Z formal notation which is supported by temporal history. We present a formal proof technique specifying and verifying that the hazardous states are not introduced into ladder logic in the PLC-based safety critical system.

• Research in Mathematical Education
• /
• v.13 no.3
• /
• pp.217-233
• /
• 2009

The purpose of this study is to describe how dynamic geometry systems can be useful in proof activity; teaching sequences based on the use of dynamic geometry systems and to analyze the possible roles of dynamic geometry systems in both teaching and learning of proof. And also dynamic geometry environments can generate powerful interplay between empirical explorations and formal proofs. The point of this study was to show that how using dynamic geometry software can provide an opportunity to link between empirical and deductive reasoning, and how such software can be utilized to gain insight into a deductive argument.

• International Journal of Railway
• /
• v.5 no.2
• /
• pp.65-70
• /
• 2012

This paper introduced a novel railway system, Automatic Train Protection and Block (ATPB) briefly, which is proposed to improve the efficiency of existing regional train lines with low cost in Japan. The biggest superiority of ATPB system is a great use of universal and mature technologies, such as GPS and regular mobile telephone networks, so that there is nearly no increment of trackside equipments in the reconstruction. Then in order to guarantee the system safety, a formal model of ATPB is established and analyzed by formal method VDM++. Firstly, the specification is specified by VDM++ formally without ambiguity. Secondly, its internal consistency is proved by discharging the proof obligations. And finally, its satisfiability is checked by systematic testing, which executes specification and checks the outputs against corresponding inputs.

• Journal of Educational Research in Mathematics
• /
• v.9 no.2
• /
• pp.419-438
• /
• 1999

This paper explored the impact of dynamic geometry software such as CabriII, GSP on student's understanding deductive justification, on the assumption that proof in school mathematics should be used in the broader, psychological sense of justification rather than in the narrow sense of deductive, formal proof. The following results have been drawn: Dynamic geometry provided positive impact on interacting between empirical justification and deductive justification, especially on understanding the necessity of deductive justification. And teacher in the computer environment played crucial role in reducing on difficulties in connecting empirical justification to deductive justification. At the beginning of the research, however, it was not the case. However, once students got intocul-de-sac in empirical justification and understood the need of deductive justification, they tried to justify deductively. Compared with current paper-and-pencil environment that many students fail to learn the basic knowledge on proof, dynamic geometry software will give more positive ffect for learning. Dynamic geometry software may promote interaction between empirical justification and edeductive justification and give a feedback to students about results of their own actions. At present, there is some very helpful computer software. However the presence of good dynamic geometry software can not be the solution in itself. Since learning on proof is a function of various factors such as curriculum organization, evaluation method, the role of teacher and student. Most of all, the meaning of proof need to be reconceptualized in the future research.

• Journal of Educational Research in Mathematics
• /
• v.21 no.1
• /
• pp.57-65
• /
• 2011

The conceptions of definition and proof radically change in the transition from secondary to tertiary mathematics. Specifically this paper analyses the historical development of the axiomatic method from Greek to modern mathematics. To understand Greek and modern axiomatic method, it is important to know the different characteristics of the primitive terms, constant and variable. Especially this matter of primitive terms explains the change of conceptions of definition, proof and mathematics. This historical analysis is useful for introducing the meaning of formal definition and proof.

• Communications of the Korean Mathematical Society
• /
• v.23 no.3
• /
• pp.307-312
• /
• 2008

In this note we first give a simple, direct proof of a combinatorial determinant involving the usual higher derivatives and then obtain a corresponding result in positive characteristic.

• International Journal of Railway
• /
• v.2 no.3
• /
• pp.99-106
• /
• 2009

Checks and tests before putting safety facilities into service as well as the results of these tests are essential, time consuming and may show great variations between each other. Economic constraints and the increasing complexity associated with the development of computerized tools tend to limit the capacity of the classic approval process (manual or automatic). A reduction of the validation cover rate could result in practice. This is not compatible with the French national plan to renew the interlocking systems of the national network. The method and the tool presented in this paper makes it possible to formally validate new computerized systems or evolutions of existing French interlocking systems with real-time functional interpreted Petri nets. The aim of our project is to provide SNCF with a method for the formal validation of French interlocking systems. A formal proof method by assertion, which is applicable to industrial automation equipment such as interlocking systems, and which covers equally the specification and its real software implementation, is presented in this paper. With the proposed method we completely verify that the system follows all safety properties at all times and does not show superfluous conditions: it replaces all the indoor checks (not the outdoor checks). The advantages expected are a significant reduction of testing time and of the related costs, an increase of the test coverage rate, an answer to the new demand of railway infrastructure maintenance engineering to modify and validate computerized interlocking systems. Formal methods mastery by infrastructure engineers are surely a key to prove that more safety is not necessarily more expensive.

• The KIPS Transactions:PartD
• /
• v.8D no.1
• /
• pp.62-70
• /
• 2001

Correct descriptions for software component functions become a strong requirement in developing critical software especially on the area of real-time applications. In this paper, we introduce both formalization of software design using patterns and verification methods in order for the components to increase their understandability. In particular, the paper investigates into a means of formal description techniques based on VDM++ for the software components, and provides adequacy proof steps for a given functional descriptions.

• Communications of Mathematical Education
• /
• v.22 no.4
• /
• pp.471-487
• /
• 2008

Inventive mathematical thinking, original mathematical problem solving ability, mathematical invention and so on are core concepts, which must be emphasized in all branches of mathematical education. In particular, Polya(1981) insisted that inventive thinking must be emphasized in a suitable level of university mathematical courses. In this paper, the author considered two cases of inventive problem solving ability shown by his many students via real analysis courses. The first case is about the proof of the problem "what is the derived set of the integers Z?" Nearly all books on mathematical analysis sent the question without the proof but some books said that the answer is "empty". Only one book written by Noh, Y. S.(2006) showed the proof by using the definition of accumulation points. But the proof process has some mistakes. But our student Kang, D. S. showed the perfect proof by using The Completeness Axiom, which is very useful in mathematical analysis. The second case is to show the infinite countability of NxN, which is shown by informal proof in many mathematical analysis books with formal proofs. Some students who argued the informal proof as an unreasonable proof were asked to join with us in finding the one-to-one correspondences between NxN and N. Many students worked hard and find two singled-valued mappings and one set-valued mapping covering eight diagrams in the paper. The problems are not easy and the proofs are a little complicated. All the proofs shown in this paper are original and right, so the proofs are deserving of inventive mathematical thoughts, original mathematical problem solving abilities and mathematical inventions. From the inventive proofs of his students, the author confirmed that any students can develope their mathematical abilities by their professors' encouragements.