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

Naming and necessity

Reasoning is generally seen as a means to improve knowledge and make better decisions. However, much evidence shows that reasoning often leads to epistemic distortions and poor decisions. This suggests that the function of reasoning should be rethought. Our hypothesis is that the function of reasoning is argumentative. It is to devise and evaluate arguments intended to persuade. Reasoning so conceived is adaptive given the exceptional dependence of humans on communication and their vulnerability to misinformation. A wide range of evidence in the psychology of reasoning and decision making can be reinterpreted and better explained in the light of this hypothesis. Poor performance in standard reasoning tasks is explained by the lack of argumentative context. When the same problems are placed in a proper argumentative setting, people turn out to be skilled arguers. Skilled arguers, however, are not after the truth but after arguments supporting their views. This explains the notorious confirmation bias. This bias is apparent not only when people are actually arguing, but also when they are reasoning proactively from the perspective of having to defend their opinions. Reasoning so motivated can distort evaluations and attitudes and allow erroneous beliefs to persist. Proactively used reasoning also favors decisions that are easy to justify but not necessarily better. In all these instances traditionally described as failures or flaws, reasoning does exactly what can be expected of an argumentative device: Look for arguments that support a given conclusion, and, ceteris paribus, favor conclusions for which arguments can be found.

/pdf/why-do-humans-reason-arguments-for-an-argumentative-theory-19genwgdwy.pdf

Why Do Humans Reason? Arguments for an Argumentative Theory

Is there a biology of intelligence which is characteristic of the normal human nervous system?" Here we review 37 modern neuroimaging studies in an attempt to address this question posed by Halstead (1947) as he and other icons of the last century endeavored to understand how brain and behavior are linked through the expression of intelligence and reason. Reviewing studies from functional (i.e., functional magnetic resonance imaging, positron emission tomography) and structural (i.e., magnetic resonance spectroscopy, diffusion tensor imaging, voxel-based morphometry) neuroimaging paradigms, we report a striking consensus suggesting that variations in a distributed network predict individual differences found on intelligence and reasoning tasks. We describe this network as the Parieto-Frontal Integration Theory (P-FIT). The P-FIT model includes, by Brodmann areas (BAs): the dorsolateral prefrontal cortex (BAs 6, 9, 10, 45, 46, 47), the inferior (BAs 39, 40) and superior (BA 7) parietal lobule, the anterior cingulate (BA 32), and regions within the temporal (BAs 21, 37) and occipital (BAs 18, 19) lobes. White matter regions (i.e., arcuate fasciculus) are also implicated. The P-FIT is examined in light of findings from human lesion studies, including missile wounds, frontal lobotomy/leukotomy, temporal lobectomy, and lesions resulting in damage to the language network (e.g., aphasia), as well as findings from imaging research identifying brain regions under significant genetic control. Overall, we conclude that modern neuroimaging techniques are beginning to articulate a biology of intelligence. We propose that the P-FIT provides a parsimonious account for many of the empirical observations, to date, which relate individual differences in intelligence test scores to variations in brain structure and function. Moreover, the model provides a framework for testing new hypotheses in future experimental designs.

/pdf/the-parieto-frontal-integration-theory-p-fit-of-intelligence-2kp9bfy1c3.pdf

The Parieto-Frontal Integration Theory (P-FIT) of intelligence: converging neuroimaging evidence.

A Study of Thinking

The authors present a similarity-based model of induction and categorization in young children (SINC). The model suggests that (a). linguistic labels contribute to the perceived similarity of compared entities and (b). categorization and induction are a function of similarity computed over perceptual information and linguistic labels. The model also predicts young children's similarity judgment, induction, and categorization performance under different stimuli and task conditions. Predictions of the model were tested and confirmed in 6 experiments, in which 4- to 5-year-olds performed similarity judgment, induction, and categorization tasks using artificial and real labels (Experiments 1-4) and recognition memory tasks (Experiments 5A and 5B). Results corroborate the similarity-based account of young children's induction and categorization, and they support both qualitative and quantitative predictions of the model.

/pdf/induction-and-categorization-in-young-children-a-similarity-4jw7h9pclq.pdf

Induction and categorization in young children: a similarity-based model.

Undergraduate students may benefit more from learning mathematics through a single abstract, symbolic representation than from learning multiple concrete examples.

http://science.sciencemag.org/content/sci/320/5875/454.full.pdf

The Advantage of Abstract Examples in Learning Math

Young children often have a preference for auditory input, with auditory input often overshadowing visual input. The current research investigated the developmental trajectory and factors underlying these effects with 137 infants, 132 four-year-olds, and 89 adults. Auditory preference reverses with age: Infants demonstrated an auditory preference, 4-year-olds switched between auditory and visual preference, and adults demonstrated a visual preference. Furthermore, younger participants were likely to process stimuli only in the preferred modality, thus exhibiting modality dominance, whereas adults processed stimuli in both modalities. Finally, younger participants ably processed stimuli presented to the nonpreferred modality when presented in isolation, indicating that auditory and visual stimuli may be competing for attention early in development. Underlying factors and broader implications of these findings are discussed.

/pdf/auditory-dominance-and-its-change-in-the-course-of-plcdeevla1.pdf

Auditory Dominance and Its Change in the Course of Development

https://faculty.psy.ohio-state.edu/sloutsky/pdf/Sloutsky_s_TICS_paper-published.pdf

The role of similarity in the development of categorization

Linguistic labels play an important role in young children's conceptual organization: When 2 entities share a label, people expect these entities to share many other properties. Two classes of explanations of the importance of labels seem plausible: a language-specific and a general auditory explanation. The general auditory explanation argues that the importance of labels stems from a privileged processing status of auditory input (as compared with visual input) for young children. This hypothesis was tested and supported in 4 experiments. When auditory and visual stimuli were presented separately, 4-year-olds were likely to process both kinds of stimuli, whereas when auditory and visual stimuli were presented simultaneously, 4-year-olds were more likely to process auditory stimuli than visual stimuli.

https://faculty.psy.ohio-state.edu/sloutsky/pdf/Sloutsky___Napolitano_cdev_570.pdf

Vladimir M. Sloutsky

Papers

Induction and categorization in young children: a similarity-based model.

The Advantage of Abstract Examples in Learning Math

Auditory Dominance and Its Change in the Course of Development

The role of similarity in the development of categorization

Is a Picture Worth a Thousand Words? Preference for Auditory Modality in Young Children