• Journal of the Korean Mathematical Society
• /
• v.54 no.6
• /
• pp.1801-1816
• /
• 2017

We study some spectral properties of Volterra-type integral operators $V_g$ and $I_g$ with holomorphic symbol g on the Fock-Sobolev spaces ${\mathcal{F}}^p_{{\psi}m}$. We showed that $V_g$ is bounded on ${\mathcal{F}}^p_{{\psi}m}$ if and only if g is a complex polynomial of degree not exceeding two, while compactness of $V_g$ is described by degree of g being not bigger than one. We also identified all those positive numbers p for which the operator $V_g$ belongs to the Schatten $S_p$ classes. Finally, we characterize the spectrum of $V_g$ in terms of a closed disk of radius twice the coefficient of the highest degree term in a polynomial expansion of g.

• Honam Mathematical Journal
• /
• v.37 no.3
• /
• pp.339-352
• /
• 2015

We aim at giving explicit expressions of $${\sum_{m,n=0}^{{\infty}}}{\frac{{\Delta}_{m+n}(-1)^nx^{m+n}}{({\rho})_m({\rho}+i)_nm!n!}$$, where i = 0, ${\pm}1$, ${\ldots}$, ${\pm}9$ and $\{{\Delta}_n\}$ is a bounded sequence of complex numbers. The main result is derived with the help of the generalized Kummer's summation theorem for the series $_2F_1$ obtained earlier by Choi. Further some special cases of the main result considered here are shown to include the results obtained earlier by Kim and Rathie and the identity due to Bailey.

• Communications of the Korean Mathematical Society
• /
• v.30 no.4
• /
• pp.439-446
• /
• 2015

The main objective of this paper is to establish certain explicit expressions for the Humbert functions ${\Phi}_2$(a, a + i ; c ; x, -x) and ${\Psi}_2$(a ; c, c + i ; x, -x) for i = 0, ${\pm}1$, ${\pm}2$, ..., ${\pm}5$. Several new and known summation formulas for ${\Phi}_2$ and ${\Psi}_2$ are considered as special cases of our main identities.

• School Mathematics
• /
• v.7 no.1
• /
• pp.55-75
• /
• 2005

Mathematical symbolizing is an important part of mathematics learning. But many students have difficulties m symbolizing mathematical ideas formally. If students had experiences inventing their own mathematical symbols and developing them to conventional ones natural way, i.e. learning mathematical symbols via expressive approaches, they could understand and use formal mathematical symbols meaningfully. These experiences are especially valuable for students who meet mathematical symbols for the first time. Hence, there are needs to investigate how early elementary school students can and should experience meaningful mathematical symbolizing. The purpose of this study was to analyze students' mathematical symbolizing processes and characteristics of theses. We carried out teaching experiments that promoted meaningful mathematical symbolizing among eight first graders. And then we analyzed students' symbolizing processes and characteristics of expressive approaches to mathematical symbols in early elementary students. As a result, we could places mathematical symbolizing processes developed in the teaching experiments under five categories. And we extracted and discussed several characteristics of early elementary students' meaningful mathematical symbolizing processes.

• Journal of the Korean School Mathematics Society
• /
• v.23 no.1
• /
• pp.67-88
• /
• 2020

This study analyzed the characteristics of prospective teachers' Horizon Content Knowledge(HCK) related to understandings of an inverse function symbol. This study aimed to deduce implications of developing HCK in terms of the means which would enhance mathematics teachers' professional development. In order to achieve the aim, this study identified features of HCK by examining the previous literature on HCK, which has conformed Ball & Bass(2009) and exploring the research in AMT, including Zazkis & Leikin(2010) which has emphasized cultivating AMT through university mathematics education. In addition, a questionnaire was developed regarding the features of HCK and taken by 57 prospective teachers. By analyzing the data obtained from the written responses the participants presented, this study delineated the specific characteristics of the teachers' HCK with regard to an inverse function symbol. Additionally, several issues in the teacher education for improving HCK were discussed, and the results of this research could inspire designing and implementing a teacher education program relevant to HCK.

• Journal of Educational Research in Mathematics
• /
• v.21 no.2
• /
• pp.165-180
• /
• 2011

Mathematical symbols concisely represent mathematical contents related to terms by describing their mathematical meanings implicitly. All symbols in elementary school mathematics textbooks are stated as to be read so that elementary school students could understand their mathematical meanings. The same is somewhat true as in middle school mathematics textbooks, however it is often the case that some symbols are difficult to be read and understood because their statements are unclear or different. In this study, we analyze problems and suggest implications on teaching and learning mathematics based on the statements and understanding of reading symbols in middle school mathematics textbooks.

• Education of Primary School Mathematics
• /
• v.7 no.2
• /
• pp.85-99
• /
• 2003

In this paper, firstly examined various proofs types that cover informal empirical justifications by Balacheff, Miyazaki, and Harel ＆ Sowder and Tall. Using these theoretical frameworks, justification activities by 5th graders were analyzed and several conclusions were drawn as follow: 1) Children in 5th grade could justify using various proofs types and method ranged from external proofs schemes by Harel & Sowder to thought experiment by Balacheff This implies that children in elementary school can justify various mathematical statements of ideas for themselves. To improve children's proving abilities, rich experience for justifying should be provided. 2) Activities that make conjectures from cases then justify should be given to students in order to develop a sense of necessity of formal proof. 3) Children have to understand the meaning and usage of mathematical symbol to advance to formal deductive proofs. 4) New theoretical framework is needed to be established to provide a framework for research on elementary school children's justification activities. Research on proof mainly focused on the type of proof in terms of reasoning and activities involved. But proof types are also influenced by the tasks given. In elementary school, tasks that require physical activities or examples are provided. To develop students'various proof types, tasks that require various justification methods should be provided. 5) Children's justification type were influenced not only by development level but also by the concept they had. 6) Justification activities provide useful situation that assess students'mathematical understanding. 7) Teachers understanding toward role of proof(verification, explanation, communication, discovery, systematization) should be the starting point of proof activities.

• Korean Journal of Mathematics
• /
• v.15 no.2
• /
• pp.185-193
• /
• 2007

In this paper we consider the hyponormality of Toeplitz operators $T_{\varphi}$ on the Bergman space $L^2_a({\mathbb{D})$ with symbol in the case of function $f+{\overline{g}}$ with polynomials $f$ and $g$. We present some necessary conditions for the hyponormality of $T_{\varphi}$ under certain assumptions about the coefficients of ${\varphi}$.

• School Mathematics
• /
• v.9 no.2
• /
• pp.181-196
• /
• 2007

It would have a trouble to communicate mathematically without an appropriate use of mathematical language. Therefore it is necessary to form mathematics classroom culture to encourage students to use mathematical language precisely. A four-month teaching experiment in a 4th grade mathematics class was conducted focused the accurate use of mathematical language. In the course of the teaching experiment, children became more careful to use their language precisely. The use of demonstrative pronouns such as this or that as well as the use of inaccurate or wrong expressions was diminished. Children became to use much more mathematical symbols and terms instead of their imprecise expressions. The result of the experiment suggests that the culture that encourage students to use mathematical language precisely can be formed in elementary mathematics classroom.

• Journal of Educational Research in Mathematics
• /
• v.13 no.2
• /
• pp.123-141
• /
• 2003

This study investigated the understanding level and the using level of mathematical language for middle school students in terms of Freudenthal' language levels. It was proved that the understanding level task developed by current study for geometric concept had reliability and validity, and that there was the hierarchy of levels on which students understanded mathematical language. The level that students used in explaining mathematical concepts was not interrelated to the understanding level, and was different from answering the right answer according to the sorts of tasks. And, the level of mathematical language that was understood easily as students' thought, was the third level of the understanding levels. Mathematics teachers should consider the students' understanding level and using level, and give students the tasks which students could use their mathematical language confidently.