곡선 아래 넓이에서 리만합의 극한으로: 수학사를 통한 2015 개정 고등학교 수학에서 대학 미적분학으로의 정적분 담론 이행

이지현 대한수학교육학회 2023 수학교육학연구 Vol.33 No.2

The transition from high school to university mathematics presents significant challenges for understanding definite integrals. In high school, students learn that definite integral is the difference between the two antiderivatives evaluated at the endpoints. However, in university calculus, the definition of the definite integral is based on the limit of Riemann sums. To facilitate students’ transition, we employed commognitive theory to design teaching and learning activities that incorporated the history of definite integrals. Our study aimed to document changes in students’ discourse on definite integrals and analyze the conflicts they encountered in the transition. In the pre-survey, most students believed that the area under a curve was the definition of the definite integral. However, in the post-survey, many students accepted the limit of Riemann sums as the formal definition. Moreover, students demonstrated growth in substantiating narratives on the connection between areas (or the limits of Riemann sums) and antiderivatives. Utilizing the history of mathematics, we provided suitable contexts to discuss explicitly the change in word use of the definite integral. Students’ self-reports indicated that history helped them understand the reasons behind the new definition of definite integral. Additionally, rather than disregarding the high school definition as incorrect or imprecise, students could relearn it in connection with the university discourse. Our study contributes to designing a smooth transition from high school to university mathematics and for mathematics teachers to adapt to the inverse transition. 2015 고등학교 교육과정의 변화로 정적분 정의가 부정적분의 차에서 리만합의 극한으로 바뀌는 고등학교 수학에서 대학수학으로의 이행은 대학미적분학 교육에 중요한 도전 과제를 제공하고 있다. 이 연구는 commognitive 이론을 바탕으로 이러한 대학수학으로의 이행에 요구되는 메타 수준변화를 촉진하기 위해 수학사에 기반한 교수 학습 활동을 설계 및 적용하였으며, 학생들의 정적분 담론 변화와 이행에서의 commognitive 갈등을분석하였다. 대학에서의 정적분 학습 전 대부분의 학생들은 정적분 정의를 ‘곡선 아래의 넓이’ 그 자체로 생각하고 있었다. 그러나 사후에는 상당수 학생들이 리만합의 극한이라는 대학수학의 정적분 정의를 수용했으며, 넓이(혹은 리만합의 극한)와 부정적분의 관계에 대한 정당화 내러티브의성장을 확인할 수 있었다. 수학사는 고등학교 수학의 ‘부정적분의 차’뿐 아니라 학생들이 정적분의 정의라고 생각했던 ‘곡선 아래의 넓이’로부터대학수학의 리만합의 극한으로의 정적분 용어 사용의 변화를 명시적으로 다룰 수 있는 맥락을 제공했다. 학생들의 수학사 학습 경험에 대한자기보고는 수학사 학습이 리만합의 극한이라는 새로운 용어 사용의 이유를 이해하고, 고등학교의 정적분 정의를 정확하지 않은 혹은 잘못된 것으로대학수학 학습 과정에서 부정하거나 회피하는 대신 대학수학과의 연속선 상에서 재 학습할 수 있는 기회를 제공했음을 보여주고 있다. 정적분에대해 commognitive 이론을 바탕으로 고등학교 수학에서 대학수학으로의 연속적인 이행을 촉진할 수 있는 방안을 설계하고 그 결과를 분석한 본 사례연구는, 수학교사들이 직면하는 대학수학에서 고등학교수학으로의 연속적인 역이행 방안을 탐색하는 데에도 기여할 수 있을 것이다.

For The Matrix Sandwich Problems

Nae-Gyeong Jeong(정내경),Se-Won Park(박세원) 한국엔터테인먼트산업학회 2015 한국엔터테인먼트산업학회논문지 Vol.9 No.2

행렬은 다양한 속성에 기초하여 여러문제로 연구되어 왔지만, 명확-긍정적 완료 문제(The positive definite completion problems)가 가장 주목 받고 있다 . 우리는 몇 가지 조건에 명확-긍정적 완료 문제를 연구하고 부분 행렬의 완료는 통상의 행렬에 결과불특정 항목에 대한 값을 선택하고, 양극 확정적(semi-definite) 완료 문제는 주어진 부분대칭 행렬의 양극 확정적(semidefinit ) 완료가 존재하는지의 여부를 결정하는 것이다 . 명확-긍정적 행렬의 부분행렬은 어떤 주체 명확한 양극이기 때문에, 대칭 행렬 부분은 양극 확실한 완성이 필요하고 조건은 한정된 양극의 부분 일수 있다는 점이다 . 이것의 명백한 필요조건은 그래프가 현모양인 경우에 충분하다. 본 논문에서는 비대칭 (0,1)-노말행렬에 집중한 샌드위치 문제의 형태를 제공한다. Problems based on a variety of properties have been studied, but the positive definite completion problem has received the most attention. We research the positive definite problem on the several conditions. A completion of a partial matrix is a choice of values for the unspecified entries resulting in a conventional matrix, and the positive definite (semidefinite) completion problem is to determine whether there exists a positive definite (semidefinite) completion of a given partial symmetric matrix. Since any principal submatrix of a positive definite matrix is positive definite, a necessary condition that a partial symmetric matrix A have a positive definite completion is that A be partial positive definite. The same is true for positive semidefinite. This obvious necessary condition is also sufficient in the case that the graph of A is chordal. In this paper we give the matrix sandwich that concentrate on nonsymmetric (0,1)-normal matrices.

접촉에서 변형으로: 불가리아어와 마케도니아어에서 후치정관사 범주의 출현과 활용을 중심으로

채유정 중앙대학교 외국학연구소 2021 외국학연구 Vol.- No.57

The Old Slavic language, which did not have a grammatical mechanism of a definite article, used forms such as common nouns, proper nouns, and denoting pronouns to express definiteness. The emergence of definite articles, and post-positional definite articles in the Slavic language is a important issue in slavic linguistic history. In the meantime, academic explanation for this is that the appearance of the post-positional definite article in Bulgarian and Macedonian was grammatically fixed in the ancient Slavic demonstrative pronoun, which was used as a substitute for the Greek pre-positional article in the translation of the Greek Bible. However, looking at this phenomenon from the sociolinguistics and typological perspective of Balkan linguistics, it is significantly important the relationship between Romanian and Illyrian linguistic traditions. The primary purpose of this paper is to study the precise and in-depth correlation between language contact and language transformation. It is a verification that the post-positional definite article form, which was not inherently, was created through linguistic contact between Balkan languages. In conclusion, post-positional definite articles in Bulgarian or Macedonian have the tangible characteristics of Slavic in terms of form, and can be said to be the result of contact with Balkan languages ​​in terms of location. Concepts and meanings related to definiteness include specificity, deixis meaning, coherence and genericity. A study on the concepts related to these definiteness can additionally explain the semantic background of the emergence of post definite articles in Bulgarian and Macedonian languages, and differentiate them from similar grammatical mechanisms such as common nouns, proper nouns, and demonstrative pronouns and their semantic meaning.

Lacroix식 정적분 정의에 기반한 적분법 구성 방식에 대한 고찰: 20세기 전반기 미적분 교재를 중심으로

정연준 대한수학교육학회 2021 학교수학 Vol.23 No.1

In the current curriculum, The result of the fundamental theorem of calculus, F(b)-F(a) is presented as the definition of definite integral. This definition of definite integral was proposed by Lacroix in the lately 18th century and was used in various calculus textbooks in the first half of the 20th century. In this study, I analyzed the relationship between the differential and the integral calculus, searched the mathematical meaning of the definition of the Lcaroix-style definition of the definite integral, and surveyed the integral calculus chapters of calculus textbooks that adopted such definition. Based on these discussions, I proposed some suggestions to improve the teaching of the integral calculus. 2015 개정 교육과정에서 정적분 정의가 로 바뀌었다. 이러한 정적분 정의는 역사적으로 18세기 말 Lacroix에 의해 제시된 바 있으며, 20세기 전반기에도 여러 미적분 교재에서 사용되었다. 이 연구에서는 Lacroix식 정적분의 수학적 타당성을 살펴보고, 적분의 두 측면에 대한 논의를 바탕으로 하여 Lcaroix식 정적분 정의의 교수학적 의미를 탐색하였다. 논의 결과를 바탕으로 하여, Lacroix식 정적분 정의를 채택한 20세기 미적분 교재를 분석하고, 적분법 지도 개선에 필요한 사항을 제시하였다.

(In)definiteness in Korean without an article

Kang, Arum,Park, Myung-Kwan 한국언어학회 2020 언어 Vol.45 No.3

Bare NPs in Korean have recently drawn increasing attention since apparently without any overt grammatical article/determiner on them, they are interpreted at least in three ways: generic, indefinite, and definite. Concentrating on the latter two readings, we investigate how bare NPs in Korean yield them. We first show that indefinites in Korean that correspond to an indefinite article headed DP in English are represented by bare NPs, though specificity-encoding and number-in-focus ones are represented by singular numeral classified NPs. We then show that definites in Korean analogous to definite article headed DPs in English are also represented by article-less bare NPs. Since like in English, in Korean there is no distinction in form between uniqueness-denoting weak definites and familiarity-denoting anaphoric strong definites, it follows that in Korean without an article, bare NPs can serve as both types of definites. Particularly, bare NPs as strong definites come about in restricted discourse contexts where two or more sentences are part of one larger topic situation. Meanwhile, personal names are referred back to by demonstrative-preceded bare NPs, which indicates that in Korean, the demonstrative is starting to develop into a definite article-like determiner.

For The Matrix Sandwich Problems

정내경,박세원 한국엔터테인먼트산업학회 2015 한국엔터테인먼트산업학회논문지 Vol.9 No.2

Problems based on a variety of properties have been studied, but the positive definite completion problem has received the most attention. We research the positive definite problem on the several conditions. A completion of a partial matrix is a choice of values for the unspecified entries resulting in a conventional matrix, and the positive definite (semidefinite) completion problem is to determine whether there exists a positive definite (semidefinite) completion of a given partial symmetric matrix. Since any principal submatrix of a positive definite matrix is positive definite, a necessary condition that a partial symmetric matrix A have a positive definite completion is that A be partial positive definite. The same is true for positive semidefinite. This obvious necessary condition is also sufficient in the case that the graph of A is chordal. In this paper we give the matrix sandwich that concentrate on nonsymmetric (0,1)-normal matrices.

스페인어 초급문법교과의 관사 용법에 대한 평가분석

곽재용(Kwak, Jae-Yong) 한국외국어교육학회 2009 Foreign languages education Vol.16 No.1

The most of Spanish Grammar textbooks generally explain the distinction between the two types of normal expressions -nouns with definite/indefinite article and bare nouns (nouns without article) - based on the use of the definite article. There are two approaches in the previous studies about the definite articles: one is based on the descriptive viewpoint (Leonetti, 1999; Padron, 1999; Solis, 1999), and the other is on the acquisition per se of the articles (Cohen, 1976; Fernández, 1997; Mayberry, 1998; Salaberry & López-Ortega, 1998; Tarone, 1988). In the descriptive approach, there are more explanations about the definite article than the indefinite article. For example, Padron(1999) stipulates that the definite article is more difficult in the acquisition among the Spanish L2 learners. We will try to show that the distinction between nouns with definite/indefinite article and bare nouns must be explained as the independent usages in Spanish, not as a variety of the use of the definite article. La mayor parte de los libros de texto de la gramática espańola explica la distinción entre los dos tripos de expresiones nominales - nombres con artículo definido/indefinido y nombres escueto (nombres sin artículi)-según el uso del artículo definido. Esisten dos aproximaciones en los previos estudios sobre los artículos defínidos: una de la perspectiva descriptiva (Leonetti, 1999; Padron, 1999; Solis, 1999), y otra de la adquicisión per se de los artículos (Cohen, 1976; Fernández, 1997; Mayberry, 1998; Salaberry & López-Ortega, 1998; Tarone, 1988). Bajo la aproximación descriptiva, se encuentran más explicaciones sobre el artículo definido que el artículo defínido resulta más difícil en la adquisición entre los estudiantes de L2 de espańol. Intentaremos mostrar que la distinción entre nombres con artículo defínido/indefínido y nombres escuetos deben explicarse como usos independientes en espańol, no como una variedad del uso del artículo defínido.

Anaphoric Definites without a Determiner: Korean vs. Chinese

( Myung-kwan Park ),( Arum Kang ) 서울대학교 인지과학연구소 2024 Journal of Cognitive Science Vol.25 No.1

This paper contributes to the recent works on the nature of anaphoric definites by comparing Korean and Mandarin Chinese (MC). We first note that in Korean, bare NPs as well as demonstrative-preceded nominals can serve as an anaphoric definite. This sharply contrasts with MC, where demonstrative-preceded nominals can do so, but bare NPs can do so only in the subject/topic position and not in other positions. Building on the claim that unlike unique definites (or weak Ds), anaphoric definites (or strong Ds) are composed of a strong determiner whose anaphoric index argument establishes an explicitly expressed relation with their antecedents (i.e., Schwarz 2009, Jenks 2018), we propose that there are two loci for structurally accommodating its anaphoric index argument: (i) at the periphery of Nume(ral)P; and (ii) at the periphery of bare NP. Both Korean and MC demonstratives just outside NumeP assume the role of a strong determiner, performing the same function as the English definite determiner the. However, Korean diverges from MC in structurally licensing the covert strong-determiner-like element just outside a bare NP. It can be lexicalized and licensed with the Case/case morphology available to Korean but only by topic features in the subject/topic position in MC. An important consequence of this analysis is that because unlike their counterparts in Korean, bare NPs in the object position in MC cannot be an anaphoric definite, neither can their null counterparts (i.e., pro's) in the object position in MC, which echoes Huang’s (1984) celebrated ban on null anaphoric definites in the object position. Demonstrating that apparently null anaphoric definites are available only in VP-ellipsis constructions in MC (as well as Korean), we argue that in Korean and MC, overt demonstrative pronouns replace demonstrative-preceded nominals, and pro's substitute bare NPs.

Comparison of English and Korean Definite/Indefinite Markers

( Ae Ryung Kim ) 한국현대영어영문학회 2011 현대영어영문학 Vol.55 No.3

This paper intends to seek a way to make English definiteness and indefiniteness easily accessible to Korean learners of English─a parallel analysis of English and Korean forms of definiteness and indefiniteness. First, it makes a fine-grained characterization of English (in)definiteness within the framework of the Givenness Hierarchy, which co-relates between cognitive statuses of referents and their forms. The forms are hierarchically entailed as follows: this, that, this N > that N > the N > indefinite-this N > a N. Then the paper pursues an analysis of Korean (in)definite markers in a parallel manner with English markers. The markers show a similar hierarchically entailed relation as follows: i, ce N > ku N > ? N > han N, etten N > ? N, N with a numeral classifier. (Kyungnam University)

An Incorporation Approach for Weak Definites

( Keun Young Shin ) 한국외국어대학교 언어연구소 2013 언어와 언어학 Vol.0 No.59

This paper discusses shortcomings with a verb shifting approach for weak definites and argues for an incorporation approach. Aguilar-Guevara and Zwarts(2010) claim that weak definites refer to kind individuals and combine with transitive verbs which are shifted to special kind-level predicates. This verb shifting approach has difficulty in capturing the fact that the semantic scope of nouns in weak definites is indeterminate and being extended to account for bare singulars which share linguistic properties with weak definites. By adopting the idea that nouns in weak definites express activities, I propose that a weak definite forms a complex predicate with a verb, sharing their arguments.