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Using Multivariate Statistics

Co-Planar Stereotaxic Atlas of the Human Brain—3-Dimensional Proportional System: An Approach to Cerebral Imaging, J. Talairach, P. Tournoux. Georg Thieme Verlag, New York (1988), 122 pp., 130 figs. DM 268

The power of feedback.

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

(1998). Human cognitive abilities: A survey of factor analytic studies. Gifted and Talented International: Vol. 13, No. 2, pp. 97-98.

Human cognitive abilities: A survey of factor analytic studies

https://www.uni-trier.de/fileadmin/fb1/prof/PSY/PAE/Forschung/SieglerEtAl2011.pdf

An integrated theory of whole number and fractions development.

The last 2 decades witnessed a surge in empirical studies on the variables associated with achievement in higher education. A number of meta-analyses synthesized these findings. In our systematic literature review, we included 38 meta-analyses investigating 105 correlates of achievement, based on 3,330 effect sizes from almost 2 million students. We provide a list of the 105 variables, ordered by the effect size, and summary statistics for central research topics. The results highlight the close relation between social interaction in courses and achievement. Achievement is also strongly associated with the stimulation of meaningful learning by presenting information in a clear way, relating it to the students, and using conceptually demanding learning tasks. Instruction and communication technology has comparably weak effect sizes, which did not increase over time. Strong moderator effects are found for almost all instructional methods, indicating that how a method is implemented in detail strongly affects achievement. Teachers with high-achieving students invest time and effort in designing the microstructure of their courses, establish clear learning goals, and employ feedback practices. This emphasizes the importance of teacher training in higher education. Students with high achievement are characterized by high self-efficacy, high prior achievement and intelligence, conscientiousness, and the goal-directed use of learning strategies. Barring the paucity of controlled experiments and the lack of meta-analyses on recent educational innovations, the variables associated with achievement in higher education are generally well investigated and well understood. By using these findings, teachers, university administrators, and policymakers can increase the effectivity of higher education. (PsycINFO Database Record

Variables associated with achievement in higher education: A systematic review of meta-analyses.

Many studies have investigated the association between numerical magnitude processing skills, as assessed by the numerical magnitude comparison task, and broader mathematical competence, e.g. counting, arithmetic, or algebra. Most correlations were positive but varied considerably in their strengths. It remains unclear whether and to what extent the strength of these associations differs systematically between non-symbolic and symbolic magnitude comparison tasks and whether age, magnitude comparison measures or mathematical competence measures are additional moderators. We investigated these questions by means of a meta-analysis. The literature search yielded 45 articles reporting 284 effect sizes found with 17,201 participants. Effect sizes were combined by means of a two-level random-effects regression model. The effect size was significantly higher for the symbolic (r = .302, 95% CI [.243, .361]) than for the non-symbolic (r = .241, 95% CI [.198, .284]) magnitude comparison task and decreased very slightly with age. The correlation was higher for solution rates and Weber fractions than for alternative measures of comparison proficiency. It was higher for mathematical competencies that rely more heavily on the processing of magnitudes (i.e. mental arithmetic and early mathematical abilities) than for others. The results support the view that magnitude processing is reliably associated with mathematical competence over the lifespan in a wide range of tasks, measures and mathematical subdomains. The association is stronger for symbolic than for non-symbolic numerical magnitude processing. So symbolic magnitude processing might be a more eligible candidate to be targeted by diagnostic screening instruments and interventions for school-aged children and for adults.

/pdf/associations-of-non-symbolic-and-symbolic-numerical-4ihgrhkq9j.pdf

Associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence: a meta-analysis

Mathematical competence rests on developing knowledge of concepts and of procedures (i.e. conceptual and procedural knowledge). Although there is some variability in how these constructs are defined and measured, there is general consensus that the relations between conceptual and procedural knowledge are often bi-directional and iterative. The chapter reviews recent studies on the relations between conceptual and procedural knowledge in mathematics and highlights examples of instructional methods for supporting both types of knowledge. It concludes with important issues to address in future research, including gathering evidence for the validity of measures of conceptual and procedural knowledge and specifying more comprehensive models for how conceptual and procedural knowledge develop over time.

/pdf/developing-conceptual-and-procedural-knowledge-of-biibuozuif.pdf

Developing Conceptual and Procedural Knowledge of Mathematics

We tested whether adults can use integrated, analog, magnitude representations to compare the values of fractions. The only previous study on this question concluded that even college students cannot form such representations and instead compare fraction magnitudes by representing numerators and denominators as separate whole numbers. However, atypical characteristics of the presented fractions might have provoked the use of atypical comparison strategies in that study. In our 3 experiments, university and community college students compared more balanced sets of single-digit and multi-digit fractions and consistently exhibited a logarithmic distance effect. Thus, adults used integrated, analog representations, akin to a mental number line, to compare fraction magnitudes. We interpret differences between the past and present findings in terms of different stimuli eliciting different solution strategies.

/pdf/representations-of-the-magnitudes-of-fractions-5agq8yz3u7.pdf

Michael Schneider

Papers

An integrated theory of whole number and fractions development.

Variables associated with achievement in higher education: A systematic review of meta-analyses.

Associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence: a meta-analysis

Developing Conceptual and Procedural Knowledge of Mathematics

Representations of the magnitudes of fractions.