(A) comparison of mathematically talented and non-talented students levels of thinking with regard to statistical variability

국문 초록 (Abstract)

통계는 수학과는 다른 영역을 탐구하며, 또한 통계적 탐구를 안내하는 통계 고유의 개념을 가지고 있다. 특히 통계적 사고는 경험적 자료를 이용한 귀납 추론에서 핵심이 되는 불확실성에 대한 직관과 비결정론적 사고방식을 요구한다. 그러나 통계 학습을 위해 수와 연산, 비와 비율 등의 기본적인 개념부터 함수와 미적분 등의 고차원적 개념까지 수학이 필수적으로 요구된다. 이러한 이유로 연구자들은 수학적 사고와 통계적 사고의 공통점과 차이점, 수학적 추론 능력과 통계적 추론 능력 사이의 관계에 주목해 왔다.
수학에서의 성취와 통계에서의 성취 사이의 관계를 조사한 일부 연구에서 비교적 강한 상관관계가 확인되었으나, 일부 연구에서는 매우 낮은 상관관계가 확인되었다. 그리고 수학적 추론 능력과 통계적 추론 능력 사이의 상관관계가 통계적 개념에 따라 다른 경향을 보임을 확인한 연구도 있다. 이와 같이 수학적 사고와 통계적 사고 사이의 관계에 대하여 상이한 연구결과가 제시되고 있으며, 또한 이에 대한 논의는 여전히 진행 중이다.
본 연구에서는 수학적 사고 능력이 서로 다른 두 집단 간의 통계적 사고 능력 비교를 통해 수학적 사고 능력과 통계적 사고 능력 사이의 관계를 조사한다. 이를 위해 일반학급 학생들과의 비교를 통해 수학영재학급 학생들의 통계적 사고 능력은 어떠한지 조사하였다. 먼저 통계적 변이성에 대한 사고 요소를 변이성 인식, 변이성 설명, 변이성 제어, 변이성 모델링, 표본의 이해, 표집분포의 이해로 구분하였다. 그리고 각 요소에 대한 학생들의 사고 수준을 조사하기 위해 사고 수준 비교틀을 개발하고, 일반학급 학생들과 수학영재학급 학생들의 통계적 변이성에 대한 사고 수준을 조사하여 t-검정을 실시하였다. 연구결과는 다음과 같다.
변이성 인식에 대한 사고는 측정상황과 우연상황 모두에서 5개의 수준으로 구분되었다. 변이성의 편재성 인식 부족(0 수준), 변이성 인정 불안정(1 수준), 하나의 실체로서 변이성 인식 부족(2 수준), 변이성을 하나의 실체로 인식(3 수준), 분포 개념의 발달(4 수준)로 구분되었다.
변이성 설명에 대한 사고는 측정상황에서 5개의 수준으로 구분되었다. 변이성의 원인 설명에 대한 이해 부족(0 수준), 원인 인식 미흡(1 수준), 물리적 원인 제시(2 수준), 설명되지 않는 원인을 변이성의 근원으로 제시(3 수준), 설명되지 않는 원인을 의사-우연변이성으로 간주(4 수준)로 구분되었다. 우연상황에서도 역시 5개의 수준으로 구분되었다. 변이성의 원인 설명에 대한 이해 부족(0 수준), 원인 인식 미흡(1 수준), 물리적 원인 제시(2 수준), 우연변이성 제시(3 수준), 분포의 원인 제시(4 수준)로 구분되었다.
변이성 제어에 대한 사고는 측정상황과 우연상황 모두에서 5개의 수준으로 구분되었다. 변이성 제어에 대한 인식 부족(0 수준), 물리적 제어 방법 미고려 그리고 통계적 방법의 부적절한 사용(1 수준), 물리적 제어 방법 미고려 그리고 통계적 방법의 적절한 사용(2 수준), 물리적 제어 방법 고려 그리고 통계적 방법의 부적절한 사용(3 수준), 물리적 제어 방법 고려 그리고 통계적 방법의 적절한 사용(4 수준)으로 구분되었다.
변이성 모델링에 대한 사고는 5개의 수준으로 구분되었다. 변이성 모델링에 대한 인식 부족(0 수준), 극단적인 값에 주목(1 수준), 퍼짐이나 다수의 위치에 주목(2 수준), 중심에 주목(3 수준), 중심과 퍼짐 모두에 주목(4 수준)으로 구분되었다.
표본의 이해에 대한 사고는 5개의 수준으로 구분되었다. 표본이 모집단의 일부분이라는 인식 부족(0 수준), 모집단의 부분집합으로 인식(1 수준), 모집단의 준비례적 축소버전으로 인식(2 수준), 편의없는 표본의 중요성 인식(3 수준), 무작위 표집의 영향 이해(4 수준)로 구분되었다.
표집분포의 이해에 대한 사고는 5개의 수준으로 구분되었다. 표집변이성 인식 부족(0 수준), 표집변이성 인정(1 수준), 표본 통계량의 퍼짐에 주목(2 수준), 표본 통계량의 중심과 퍼짐에 주목(3 수준), 표본 크기와 표집변이성의 관계 인식(4 수준)으로 구분되었다.
초등학교 수학영재학급 학생들과 일반학급 학생들의 사고 수준에 대한 t-검정 결과, 측정상황에서 변이성 설명, 측정상황에서 변이성 제어, 우연상황에서 변이성 제어, 변이성 모델링, 표본의 이해, 표집분포의 이해에 대한 사고에서 두 그룹 사이에 통계적으로 유의한 차이가 있는 것으로 나타났다. 그러나 측정상황과 우연상황에서 변이성 인식, 우연상황에서 변이성 설명에 대한 사고에서 통계적으로 유의한 차이가 없는 것으로 나타났다. 그리고 각 요소별 수학영재학급 학생들의 사고 수준의 분포는 일반학급 학생들의 사고 수준의 분포와 상당 부분 중첩되며 일부 요소에서는 동일한 범위에 분포되어 있는 것으로 나타났다.
중학교 수학영재학급 학생들과 일반학급 학생들의 사고 수준에 대한 t-검정 결과, 측정상황에서 변이성 인식, 우연상황에서 변이성 인식, 측정상황에서 변이성 설명, 측정상황에서 변이성 제어, 우연상황에서 변이성 제어, 변이성 모델링, 표본의 이해에 대한 사고에서 통계적으로 유의한 차이가 있는 것으로 나타났다. 그러나, 우연상황에서 변이성 설명과 표집분포의 이해에 대한 사고에서 통계적으로 유의한 차이가 없는 것으로 나타났다. 그리고 초등학생들과 마찬가지로 각 요소별 수학영재학급 학생들의 사고 수준의 분포는 일반학급 학생들의 사고 수준의 분포와 상당 부분 중첩되며 일부 요소에서는 동일한 범위에 분포되어 있는 것으로 나타났다.
초등학생과 중학생 모두 통계적 변이성에 대한 일부 사고 요소에서 수학영재학급 학생들과 일반학급 학생들의 사고 수준 사이에 통계적으로 유의한 차이가 있는 것으로 나타난 반면, 일부 요소에서는 그렇지 않은 것으로 나타났다. 그리고 수학영재학급 학생들의 통계적 사고 수준의 분포는 일반학급 학생들의 사고 수준의 분포와 상당 부분 중첩되며 일부 요소에서는 동일한 범위에 분포되어 있는 것으로 나타났다. 오랫동안 축척된 연구 결과들은 수학영재학급 학생들이 수학적 사고 능력에서 일반학급 학생들보다 상당히 우수하다는 것을 잘 보여준다. 그러나 일반학급 학생들과의 비교를 통해 수학영재학급 학생들의 통계적 사고 수준을 조사한 본 연구의 결과는 통계적 사고에서도 역시 그러한 경향이 나타난다고 확신하기는 어렵다는 것을 보여준다. 이러한 결과는 수학적 사고 능력과 통계적 사고 능력 사이의 관계가 불분명함을 보여주는 증거가 될 수 있다.

다국어 초록 (Multilingual Abstract)

Statistics studies areas apart from mathematics and has its own unique concepts to guide statistical exploration. In particular, statistics requires a sound mind-set and acknowledgment of uncertainty which are crucial for inductive reasoning about empirical data. Although statistics is a separate field on its own, it works in tandem with mathematics; thus research has often compared statistics and mathematics in various respects and examined the relationship between students’ performance in both fields.
Studies examining the relationship between students’ performance in mathematics and statistics present a variety of opinions. For instance, some studies identified a strong relationship between students’ performance in the two fields while others found a very weak relationship. Still, others argued that the relationship between the performances in these two fields depends on the students’ grasp of statistical concepts or ideas that were selected to measure their statistical aptitude. In other words, the relationship between statistical and mathematical ability is at best inconclusive and studies regarding this issue are still ongoing. This study aims to shed light on this issue by examining the statistical abilities of mathematically talented students.
This study investigates the relationship between statistical and mathematical ability by comparing cognitive abilities of mathematically talented students in statistics with those of non-talented students. Because there is no conceptual framework suitable for examining hierarchical cognitive levels for statistical abilities, based on the literature review, this study first decomposes statistical variability thinking into six components: the noticing of variability, the explaining of variability, the controlling for variability, the modeling of variability, understanding of sample, and understanding of sampling distribution. This study then develops frameworks of hierarchical cognitive levels for comparing the thinking levels of mathematically talented and non-talented students of statistical variability. After analyzing the mathematically talented students’ thinking levels of statistical variability, this study compares them with those of the non-talented students.
The finding of this study are outlined below:
The framework for students’ noticing of variability consists of five hierarchical cognitive levels in both the measurement and chance settings: unawareness of the omnipresence of variability (level 0), inconsistent unawareness of variability (level 1), no recognition of variability as an entity (level 2), consideration of variability as an entity (level 3), and development of the concept of distribution (level 4).
The framework for students’ explaining of variability consists of five hierarchical cognitive levels: in the measurement setting, no awareness of the causes (level 0), insufficient understanding of the causes (level 1), consideration of physical causes (level 2), consideration of unexplained causes as the source of variability (level 3), and consideration of unexplained causes as quasi-chance variability (level 4); in the chance setting, lack of awareness of the causes (level 0), insufficient understanding of the causes (level 1), consideration of physical causes (level 2), recognition of chance variability (level 3), and consideration of the causes of distribution (level 4).
The framework for students’ controlling for variability consists of five hierarchical cognitive levels in both the measurement and chance settings: lack of awareness of control for variability (level 0), no consideration of physical control and inappropriate use of statistical control (level 1), no consideration of physical control and appropriate use of statistical control (level 2), consideration of physical control and inappropriate use of statistical control (level 3), and consideration of physical control and appropriate use of statistical control (level 4).
The framework for students’ modeling of variability consists of five hierarchical cognitive levels: no data-based decision (level 0), extreme-value-based decision (level 1), spread-based decision (level 2), center-based decision (level 3), and distribution-based decision (level 4).
The framework for students’ understanding of sample consists of five hierarchical cognitive levels: no recognition sample of a part of the population (level 0), consideration of samples as subsets of the population (level 1), consideration of samples as a quasi-proportional, small-scale version of the population (level 2), recognition of the importance of unbiased samples (level 3), and recognition of the effect of random sampling on samples (level 4).
The framework for students’ understanding of sampling distribution consists of five hierarchical cognitive levels: lack of recognition of sampling variability (level 0), confusion of data in a sample with sample statistics (level 1), focus on spread of sample statistics (level 2), focus on the distribution of sample statistics (level 3), recognition of the relationship between sample size and sampling variability (level 4).
Results of the comparison of mathematically talented and non-talented fifth graders using t-tests show a statistically significant difference between the two groups in their ability to explain variability in the measurement setting, control for variability both in the measurement and chance settings, model variability, their understanding of sample, and understanding of sampling distribution. However, no statistically significant difference was found between the two groups in their noticing of variability both in the measurement and chance settings and their ability to explain variability in the chance setting. The distributions of thinking levels of these two groups of students overlap extensively in each component. For some components, the distributions coincided with each other (e.g., Noticing (C), Explaining (C), and Controlling (C)). For other components, some talented students performed lower than their non-talented counterparts (e.g., Modeling).
Results of the comparison of mathematically talented and non-talented eighth graders using t-tests reveal a statistically significant difference between the two groups in their noticing of variability both in the measurement and chance settings, their ability to explain variability in the measurement setting, control for variability both in the measurement and chance settings, model variability, and in their understanding of sample. However, no statistically significant difference was found between the two groups in their ability to explain variability in the chance setting and in their understanding of sampling distribution. Just as that for the elementary students, the distributions of thinking levels of these two groups of secondary students overlap extensively in each component. For some components, the distributions coincided with each other (e.g., Controlling (C) and Modeling (C)). For other components, some talented students performed lower than the non-talented students (e.g., Sampling Distribution).
These results imply that it is difficult to say that the statistical abilities of mathematically talented students are at the same levels as their mathematical abilities when compared with non-talented students. In other words, although mathematically talented students are better than non-talented students in terms of mathematical ability in general, their statistical abilities did not exhibit the same pattern. The statistical abilities of mathematically talented students overlap with those of non-talented students. Some talented students performed better than non-talented students in some components, some performed the same as the non-talented in some components, and still some performed lower than the non-talented in other components. These results can be evidence of unclear relationship between mathematical and statistical abilities.

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