• The Mathematical Education
• /
• v.49 no.4
• /
• pp.437-462
• /
• 2010

The purpose of this research is to analyze the textbook contents about the natural number concepts in the Korean National Elementary Mathematics Curriculum. Understanding a concept of natural number is crucial in school mathematics curriculum planning, since elementary students start their basic learning with natural number system. The concepts of natural number have various meaning from the perspectives of pedagogical research, and the philosophy of mathematics. The natural number concepts in the elementary math curriculum consist of four aspects; counting numbers, cardinal numbers, ordinal numbers, and measuring numbers. Two research questions are addressed; (1) How are the natural number concepts focusing on counting, cardinal, ordinal, measuring numbers are covered in the national math curriculum? ; (2) What suggestions can be made to enhance the teaching and learning about the natural number concepts? Findings reveal that (1) the national mathematics curriculum properly reflects four aspects of natural number concepts, as the curriculum covers 50% of the cardinal number system; (2) In the aspect of the counting number, we hope to add the meaning about 'one, two, three, ......, and so on' in the Korean Mathematics curriculum. In the ordinal number, we want to be rich the related meaning in a set. Further suggestions are made for future research to include them ensuing number in the curriculum.

• Research in Mathematical Education
• /
• v.18 no.3
• /
• pp.171-185
• /
• 2014

Mathematically deductive reasoning skill is one of the major learning objectives stated in senior secondary curriculum (CDC & HKEAA, 2007, page 15). Ironically, student performance during routine assessments on geometric reasoning, such as proving geometric propositions and justifying geometric properties, is far below teacher expectations. One might argue that this is caused by teachers' lack of relevant subject content knowledge. However, recent research findings have revealed that teachers' knowledge of teaching (e.g., Ball et al., 2009) and their deductive reasoning skills also play a crucial role in student learning. Prior to a comprehensive investigation on teacher competency, we use a case study to investigate teachers' knowledge competency on how to teach their students to mathematically argue that, for example, two triangles are congruent. Deductive reasoning skill is essential to geometry. The initial findings indicate that both subject and pedagogical content knowledge are essential for effectively teaching this challenging topic. We conclude our study by suggesting a method that teachers can use to further improve their teaching effectiveness.

• Bulletin of the Korean Mathematical Society
• /
• v.58 no.5
• /
• pp.1069-1078
• /
• 2021

Let R be a commutative ring with identity. A proper ideal I of R is called 1-absorbing primary ([4]) if for all nonunit a, b, c ∈ R such that abc ∈ I, then either ab ∈ I or c ∈ ${\sqrt{1}}$. The concept of 1-absorbing primary ideals in a polynomial ring, in a PID and in idealization of a module is studied. Moreover, we introduce weakly 1-absorbing primary ideals which are generalization of weakly prime ideals and 1-absorbing primary ideals. A proper ideal I of R is called weakly 1-absorbing primary if for all nonunit a, b, c ∈ R such that 0 ≠ abc ∈ I, then either ab ∈ I or c ∈ ${\sqrt{1}}$. Some properties of weakly 1-absorbing primary ideals are investigated. For instance, weakly 1-absorbing primary ideals in decomposable rings are characterized. Among other things, it is proved that if I is a weakly 1-absorbing primary ideal of a ring R and 0 ≠ I₁I₂I₃ ⊆ I for some ideals I₁, I₂, I₃ of R such that I is free triple-zero with respect to I₁I₂I₃, then I₁I₂ ⊆ I or I₃ ⊆ I.

• Communications of the Korean Mathematical Society
• /
• v.36 no.3
• /
• pp.423-445
• /
• 2021

In this paper, a new form of poly-Genocchi polynomials is defined by means of polylogarithm, namely, the Apostol-type poly-Genocchi polynomials of higher order with parameters a, b and c. Several properties of these polynomials are established including some recurrence relations and explicit formulas, which are used to express these higher order Apostol-type poly-Genocchi polynomials in terms of Stirling numbers of the second kind, Apostol-type Bernoulli and Frobenius polynomials of higher order. Moreover, certain differential identity is obtained that leads this new form of poly-Genocchi polynomials to be classified as Appell polynomials and, consequently, draw more properties using some theorems on Appell polynomials. Furthermore, a symmetrized generalization of this new form of poly-Genocchi polynomials that possesses a double generating function is introduced. Finally, the type 2 Apostolpoly-Genocchi polynomials with parameters a, b and c are defined using the concept of polyexponential function and several identities are derived, two of which show the connections of these polynomials with Stirling numbers of the first kind and the type 2 Apostol-type poly-Bernoulli polynomials.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.4
• /
• pp.1027-1039
• /
• 2019

The concept of jump concerns the distribution of $Tur{\acute{a}}n$ densities. A number ${\alpha}\;{\in}\;[0,1)$ is a jump for r if there exists a constant c > 0 such that if the $Tur{\acute{a}}n$ density of a family $\mathfrak{F}$ of r-uniform graphs is greater than ${\alpha}$, then the $Tur{\acute{a}}n$ density of $\mathfrak{F}$ is at least ${\alpha}+c$. To determine whether a number is a jump or non-jump has been a challenging problem in extremal hypergraph theory. In this paper, we give a way to generate non-jumps for hypergraphs. We show that if ${\alpha}$, ${\beta}$ are non-jumps for $r_1$, $r_2{\geq}2$ respectively, then $\frac{{\alpha}{\beta}(r_1+r_2)!r_1^{r_1}r_2^{r_2}}{r_1!r_2!(r_1+R_2)^{r_1+r_2}}$ is a non-jump for $r_1+r_2$. We also apply the Lagrangian method to determine the $Tur{\acute{a}}n$ density of the extension of the (r - 3)-fold enlargement of a 3-uniform matching.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.5
• /
• pp.1187-1198
• /
• 2019

Let R be a domain with its field Q of quotients. An R-module M is said to be weak w-projective if $Ext^1_R(M,N)=0$ for all $N{\in}{\mathcal{P}}^{\dagger}_w$, where ${\mathcal{P}}^{\dagger}_w$ denotes the class of GV-torsionfree R-modules N with the property that $Ext^k_R(M,N)=0$ for all w-projective R-modules M and for all integers $k{\geq}1$. In this paper, we define a domain R to be w-Matlis if the weak w-projective dimension of the R-module Q is ${\leq}1$. To characterize w-Matlis domains, we introduce the concept of w-Matlis cotorsion modules and study some basic properties of w-Matlis modules. Using these concepts, we show that R is a w-Matlis domain if and only if $Ext^k_R(Q,D)=0$ for any ${\mathcal{P}}^{\dagger}_w$-divisible R-module D and any integer $k{\geq}1$, if and only if every ${\mathcal{P}}^{\dagger}_w$-divisible module is w-Matlis cotorsion, if and only if w.w-pdRQ/$R{\leq}1$.

• The Mathematical Education
• /
• v.58 no.1
• /
• pp.79-99
• /
• 2019

This study is a study to collect information about 'Limitations of functions' related learning. Especially, this study was conducted on three students who can find answers by algebraic procedure in the process of extreme problem solving. Students have had the experience of converting from their algebraic procedures to graphical expressions. This shows how they reflect on their algebraic procedures. This study is a study that observes these parts. To accomplish this, twelfth were teaching experiment in three high school students. And we analyzed the contents related to the research topic of this study. Through this, students showed the difference of expressions in the method of finding limits by using algebraic interpretation methods and graphs. In addition, we examined the connectivity of the limitations of functions problem solving process of functions using algebraic procedures and graphs in the process of converting algebraic expressions to graph expressions. This study is a study of how students construct limit concepts. As in this study, it is meaningful to accumulate practical information about students' limit conceptual composition. We hope that this study will help students to study limit concept development process for students who have no limit learning experience in the future.

• Journal for History of Mathematics
• /
• v.33 no.6
• /
• pp.327-343
• /
• 2020

This study investigate philosophical discussions related to the Zeno's paradoxes in order to derive the mathematics educational implications. The paradox of Zeno's motion is sometimes explained by the calculus theories. However, various philosophical discussions show that the resolution of Zeno's paradox by calculus is not a real solution, and the concept of a continuum which is composed of points and the real number continuum may not coincide with the physical space and time. This is supported by the fact that the hyperreal number system of nonstandard analysis could be another model of a straight line or time and that an alternative explanation of Zeno's paradox was possible by the hyperreal number system. The existence of two different theories of the continuum suggests that teachers and students may not have the same view of the continuum. It is also suggested that the real world model used in school mathematics may not necessarily match the student's intuition or mathematical practice, and that the real world application of mathematics theory should be emphasized in education as a kind of 'correspondence.'

• Journal of the Korean Mathematical Society
• /
• v.59 no.2
• /
• pp.353-365
• /
• 2022

Let 𝓐 be the collection of analytic functions f defined in 𝔻 := {ξ ∈ ℂ : |ξ| < 1} such that f(0) = f'(0) - 1 = 0. Using the concept of subordination (≺), we define $$S^*_{\ell}\;:=\;\{f{\in}A:\;\frac{{\xi}f^{\prime}({\xi})}{f({\xi})}{\prec}{\Phi}_{\ell}(\xi)=1+{\sqrt{2}{\xi}}+{\frac{{\xi}^2}{2}},\;{\xi}{\in}{\mathbb{D}}\}$$, where the function 𝚽_ℓ(ξ) maps 𝔻 univalently onto the region Ω_ℓ bounded by the limacon curve (9u² + 9v² - 18u + 5)² - 16(9u² + 9v² - 6u + 1) = 0. For 0 < r < 1, let 𝔻_r := {ξ ∈ ℂ : |ξ| < r} and 𝒢 be some geometrically defined subfamily of 𝓐. In this paper, we find the largest number 𝜌 ∈ (0, 1) and some function f₀ ∈ 𝒢 such that for each f ∈ 𝒢 𝓛_f (𝔻_r) ⊂ Ω_ℓ for every 0 < r ≤ 𝜌, and $${\mathcal{L} _{f_0}}({\partial}{\mathbb{D}_{\rho})\;{\cap}\;{\partial}{\Omega}_{\ell}\;{\not=}\;{\emptyset}$$, where the function 𝓛_f : 𝔻 → ℂ is given by $${\mathcal{L}}_f({\xi})\;:=\;{\frac{{\xi}f^{\prime}(\xi)}{f(\xi)}},\;f{\in}{\mathcal{A}}$$. Moreover, certain graphical illustrations are provided in support of the results discussed in this paper.

• Proceedings of the Korea Inteligent Information System Society Conference
• /
• 2001.01a
• /
• pp.410-420
• /
• 2001

A primitive conceptualization is defined as the set of all intended situations. A non-primitive conceptualization is defined as the set of all the pairs every of which consists of an intended knowledge system and the set of all the situations admitted by the knowledge system. The reality of a domain is considered as the set of all the situation which have ever taken place in the past, are taking place now and will take place in the future. A conceptualization is defined as precise if the set of intended situations is equal to the domain reality. The representation of various elements of a domain ontology in a model of the ontology is considered. These elements are terms for situation description and situations themselves, terms for knowledge description and knowledge systems themselves, mathematical terms and constructions, auxiliary terms and ontological agreements. It has been shown that any ontology representing a conceptualization has to be non-primitive if either (1) a conceptualization contains intended situations of different structures, or (2) a conceptualization contains concepts designated by terms for knowledge description, or (3) a conceptualization contains concept classes and determines properties of the concepts belonging to these classes, but the concepts themselves are introduced by domain knowledge, or (4) some restrictions on meanings of terms for situation description in a conceptualization depend on the meaning of terms for knowledge description.