• School Mathematics
• /
• v.12 no.4
• /
• pp.619-638
• /
• 2010

In this paper, we listed and reviewed some properties on polygonal numbers, pyramidal numbers and Pascal's triangle, and Fibonacci sequence. We discussed that the properties of gnomonic numbers, polygonal numbers and pyramidal numbers are explained integratively by introducing the generalized k-dimensional pyramidal numbers. And we also discussed that the properties of those numbers and relationships among generalized k-dimensional pyramidal numbers, Pascal's triangle and Fibonacci sequence are suitable for teaching and learning of mathematical reasoning and connections.

• The Pure and Applied Mathematics
• /
• v.6 no.2
• /
• pp.121-127
• /
• 1999

We can get the Pascal's matrix of order n by taking the first n rows of Pascal's triangle and filling in with 0's on the right. In this paper we obtain some well known combinatorial identities and a factorization of the Stirling matrix from the Pascal's matrix.

• East Asian mathematical journal
• /
• v.40 no.2
• /
• pp.209-229
• /
• 2024

In this study, we design a teaching unit that constructs Pascal graphs and extended Pascal triangles to explore number patterns inherent in them. This teaching unit is designed to consider the diachronic process of teaching-learning by combining Dörfler's theoretical generalization model with Wittmann's design science ideas, which are applied to the didactical practice of mathematization. In the teaching unit, considering the teaching-learning level of prospective teachers who studied discrete mathematics, we generalize the well-known Pascal triangle and its number patterns to extended Pascal triangles which have directed graphs(called Pascal graphs) as geometric models. In this process, the use of symbols and the introduction of variables are exhibited as important means of generalization. It provides practical experiences of mathematization to prospective teachers by going through various steps of the generalization process targeting symbols. This study reflects Wittmann's intention in that well-understood mathematics and the context of the first type of empirical research as structure-genetic didactical analysis are considered in the design of the learning environment.

• Kyungpook Mathematical Journal
• /
• v.48 no.2
• /
• pp.331-335
• /
• 2008

In this paper, we investigate the number of ones in rows of Pascal's Rectangle. Using these results, we determine the existence of regular bipartite Steinhaus graphs. Also, we give an upper bound for the minimum degree of bipartite Steinhaus graphs.

• Communications of the Korean Mathematical Society
• /
• v.17 no.3
• /
• pp.383-387
• /
• 2002

Let p be an odd prime, and let f($\varkappa$) be the interpolating polynomial associated with a table of data points (j+1, (equation omitted) ) for 0$\leq$j$\leq$p. In this article, we find congruence identities modulo p of (p-1)!f($\varkappa$), (p-2)!f($\varkappa$), and (p-3)!f($\varkappa$). Moreover we present some conjectures of these types.

• Journal of Educational Research in Mathematics
• /
• v.13 no.4
• /
• pp.495-506
• /
• 2003

In this study, 1 investigated some properties on the special hyper-dimensional figures made by the principle of the performance of equivalent forms representation. I supposed 2 definitions on the making n-dimensional figure : a cone type(hypercube) and a pillar type(simplex). We can explain that there exists only 6 4-dimensional regular polytopes as there exists only 5 regular polygons. And there are many hyper-dimensional figures, they all have sufficient condition to show the general Euler' Characteristics. And especially, we could certificate that the simplest cone type and pillar types are fitted to Pascal's Triangle and Hasse's Diagram, each other.

• Journal of the Korea Convergence Society
• /
• v.8 no.7
• /
• pp.15-21
• /
• 2017

Recently, users' interest in IoT related products is increasing as the 4th industrial revolution has become social. The types and functions of sensors used in IoT devices are becoming increasingly diverse, and mutual authentication technology of IoT devices is required. In this paper, we propose an efficient double signature authentication scheme using Pascal's triangle theory so that different types of IoT devices can operate smoothly with each other. The proposed scheme divides the authentication path between IoT devices into two (main path and auxiliary path) to guarantee authentication and integrity of the IoT device. In addition, the proposed scheme is suitable for IoT devices that require a small capacity because they generate keys so that additional encryption algorithms are unnecessary when authenticating IoT devices. As a result of the performance evaluation, the delay time of the IoT device is improved by 6.9% and the overhead is 11.1% lower than that of the existing technique. The throughput of IoT devices was improved by an average of 12.5% over the existing techniques.

• Journal of Educational Research in Mathematics
• /
• v.12 no.4
• /
• pp.509-521
• /
• 2002

The purpose of this paper is as follows: $^{\circleda}$ to disclose the essence of symmetry $^{\circledb}$ to propose the desirable strategy of problem-solving as to symmetry $^{\circledc}$ to clarify the relationship between symmetry and group $^{\circledd}$ to propose a way of introduction of 'group' in school mathematics according to its fundamental characteristic, symmetry. This study shows that the nature of symmetry is 'invariance under a transformation' and symmetry is the main idea of 'group'. In mathematics textbooks and mathematics education literature, we find out that the logic of symmetry is widespread. We illustrate two paradigmatic problem related to symmetrical logic and exemplify a desirable instruction of Pascal's triangle. This study also suggests a possibility of developing students' unformal and unconscious conception of group with sym metry idea from elementary to secondary school mathematics.

• Journal for History of Mathematics
• /
• v.20 no.4
• /
• pp.61-70
• /
• 2007

Combinatorics, often called the 21 st century mathematics, has turned out a very important subject for the present information era. Modern combinatorics has started from some mathematical works, for example, Pascal's triangle and the binomial coefficients, and Euler's problems on the partitions of integers and Konigsberg's bridge problem, and so on. In this paper, we investigate the origin of combinatorics by looking over some interesting ancient combinatorial problems and some important problems which have started various subfields of combinatorics. We also discuss a little on the role of combinatorics in mathematics and mathematics education.

• Education of Primary School Mathematics
• /
• v.9 no.1 s.17
• /
• pp.31-41
• /
• 2005

In modern theory, creativity is an aim of mathematics education not only for the gifted but also fur the general students. The assertion that we must cultivate the creativity for the gifted students and drill the mechanical activity for the general students are unreasonable. Freudenthal has advocated the reinvention method, a pedagogical principle in mathematics education, which would promote the creativity. In this method, the pupils start with a meaningful context, not ready-made concepts, and invent informative method through which he could arrive at the formative concepts progressively. In many face the reinvention method is contrary to the traditional method. In traditional method, which was named as 'concretization method' by Freudenthal, the pupils start with ready-made concepts, and applicate this concepts to various instances through which he could arrive at the understanding progressively. Freudenthal believed that the mathematical creativity could not be cultivated through the concretization method in which the teacher transmit a ready-made concept to the pupils. In the article, we close examined the reinvention method, and presented a context of delivery route which is a illustration of reinvention method. Through that context, the principle of pascal's triangle is reinvented progressively.