미국의 “전국 수학 교사 협의회”(National Council of Teachers of Mathematics, NCTM)는 1989년부터 〈학교 수학의 교육과정과 평가 규준〉(1989), 〈수학 가르침(교수)의 전문성 규준〉(1991), 〈학교 수학의 평가(시험) 규준〉(NCTM, 1995), 〈학교 수학의 원리와 규준〉(2000)을 출판하여 미국의 수학 교육 의 전망(목표, 나아갈 길)과 규준(실행 지침)을 제시하였다. 수학 교사들로 구성된 미국의 NCTM은 학생, 학부모, 학교 행정가 등 많은 사람들과 힘을 합하여 모든 학생들에게 수준 높은 수학 교육을 받을 수 있는 여건(환경, 기회)을 조성하는 데 구심점의 역할을 하였다. 한편 많은 관련 단체들은 여러 배경과 능력을 가진 학생들이 전문성을 지닌 교사(특수 교사를 일컫는 밀이 아니다. 수학 교과를 이해하고 수학의 전문성과 특수성을 가르칠 수 있는 일반 교사를 일컫는 말이다.)로부터 미래를 대비해 평등하고, 진취적이며, 지원이 잘 이루어지고, 공학 도구(IT)가 잘 갖춰진 환경에서 중요한 수학적 아이디어를 이해하면서 학습할 수 있는 수학 교실(미국에서는 우리나라처럼 수학 교사가 수학 시간에 학생의 방(교실: Homeroom)에 찾아가지 않고 학생들이 선생의 방(수학 교실: Classroom)을 찾아온다. 전형적인 수학 교실의 사진은 2쪽에 나와 있다.)을 만들기 위해 함께 힘썼다. NCTM에서 출간한 여러 규준들은 우리나라의 제6차와 제7차 교육과정에도 큰 영향을 미쳤다. 이 글에서는 NCTM(2000)에서 제시한 학습 원리를 간단히 살펴본 다음 이를 중심으로 현재 미국 수학 교육의 교수ㆍ학습 이론의 동향을 살펴본다.

미국 NCTM의 Principles and Standards for School Mathematics에 나타난 수학과 교수,학습의 이론

The SAGE Handbook of Qualitative Research

Lectures on partial differential equations

Resolving Social Conflicts: Selected Papers on Group Dynamics.@@@The More Perfect Union: A Program for the Control of Inter-Group Discrimination in the United States.

Algebra has long been identified as an area of difficulty in the teaching and learning of mathematics. Evidence of this difficulty can be found in Irish secondary-level classrooms. Chief Examiner Reports have consistently identified algebra as an area of student weakness in State examinations. In light of poor student performance, and as part of a nationwide reform of secondary mathematics curricula, a functions-based approach to teaching algebra has been adopted in Irish schools. It was introduced in September 2011 in place of the transformational (rule and procedure)-based approach which was previously used. Through comparing the diagnostic test scores of incoming students in an Irish university in the years before and after the reform, this study finds that the reformed approach has coincided with a decline in students’ technical algebraic skills. However, interviews with practising mathematics teachers reveal that this decline is not a direct result of the functions-based approach, but rather ...

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Curriculum reform in Irish secondary schools: A focus on algebra

We present a mode-sum regularization prescription for computing the vacuum polarization of a scalar field in static spherically symmetric black hole spacetimes in even dimensions. This is the first general and systematic approach to regularized vacuum polarization in higher even dimensions, building upon a previous scheme we developed for odd dimensions. Things are more complicated here since the even-dimensional propagator possesses logarithmic singularities which must be regularized. However, in spite of this complication, the regularization parameters can be computed in closed form in arbitrary even dimensions and for arbitrary metric function $f(r)$. As an explicit example of our method, we show plots for vacuum polarization of a massless scalar field in the Schwarzschild-Tangherlini spacetime for even $d=4,\dots{},10$. However, the method presented applies straightforwardly to massive fields or to nonvacuum spacetimes.

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Mode-Sum Prescription for Vacuum Polarization in Black Hole Spacetimes in Even Dimensions

This paper presents a new approach to computing the vacuum polarization of a scalar field in static, spherically symmetric backgrounds in odd dimensions. The novel approach is in expanding the two point function in Fourier modes, thereby enabling a mode by mode subtraction of infinities and a rapid convergence of the series.

Mode-Sum Prescription for the Vacuum Polarization in odd Dimensions

We consider a quantum field which is in a Hartle-Hawking state propagating in a spherically symmetric black hole space-time. We calculate the components of the stress tensor, renormalized using the Hadamard form of the Green's function, in the exterior region of this space-time. We then specialize these results to the case of the ``lukewarm'' Riessner-Nordstrom-de Sitter black hole.

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Hadamard Renormalization of the Stress Energy Tensor in a Spherically Symmetric Black Hole Space-Time with an Application to Lukewarm Black Holes

We consider a quantum field on a lukewarm black hole spacetime. We introduce a new uniform approximation to the radial equation, constructed using an extension of Green-Liouville asymptotics. We then use this new approximation to construct the renormalized vacuum polarization in the Hartle-Hawking vacuum. Previous calculations of the vacuum polarization rely on the WKB approximation to the solutions of the radial equation; however, the nonuniformity of the WKB approximations obscures the results of these calculations near both horizons. The use of our new approximation eliminates these obscurities, enabling us to obtain explicitly finite and easily calculable values of the vacuum polarization on the two horizons.

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Extended Green-Liouville asymptotics and vacuum polarization for lukewarm black holes

This paper details an initial analysis of the transition of a second level curriculum reform to higher education in Ireland. The reform entitled ‘Project Maths’ involved changes to what second level students learn in mathematics, how they learn it and how they are assessed. Changes were rolled out nationally on a phased basis in September 2010. Students who were taught and assessed through the new curriculum first entered third level education in September 2012. It is important that third level mathematics lecturers are aware of the changes to the curriculum since certain topics such as vectors and matrices are no longer taught at second level. Hence third level courses may need to be adapted accordingly. This study investigates mathematics lecturers’ awareness of Project Maths and whether they have made any adaptions to their course content, teaching and assessment approaches as a result of the new curriculum being introduced. The findings, from a return rate of 23% of eligible respondents, show that although many lecturers are mindful of the concept of Project Maths, they are not aware of the changes in full and how it affects their own course content, teaching and assessment strategies. Accordingly, the gap between second and third level education remains. This study highlights that more needs to be done to ensure there is coherent and uniform approaches to the teaching, learning and assessment of mathematics in the transition from second to third level education.

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Cormac Breen

Papers

Mode-Sum Prescription for Vacuum Polarization in Black Hole Spacetimes in Even Dimensions

Mode-Sum Prescription for the Vacuum Polarization in odd Dimensions

Hadamard Renormalization of the Stress Energy Tensor in a Spherically Symmetric Black Hole Space-Time with an Application to Lukewarm Black Holes

Extended Green-Liouville asymptotics and vacuum polarization for lukewarm black holes

Mind the Gap: An Initial Analysis of the Transition of a Second level Curriculum Reform to Higher Education