中学数学部分概率内容的教学策略
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摘要
针对超几何分布、条件概率、离散型随机变量与分布列、相互独立的随机事件与二项分布、数学期望与方差以及正态分布等中学数学概率中几个重要概念,创设了真实的问题情境引导课堂教学,为教师的实际教学提供了具有可操作性的教学方案.特别对于正态分布密度函数的处理既不同于大学教材中的公式化定义,也不同于中学教材中频率直方图的极限定义.公式化的定义对于中学生显得有些抽象,但利用频率直方图的极限定义,正态密度函数超出了中学生的知识点和认知能力,实际教学的可操作性不大.首先,从函数模拟现实图象出发,通过现实的图形与频率直方图的相似性找出模拟这类图形的函数;其次,讨论这类函数中的参数与图象之间的关系;最后,再回到概率,讨论参数与数学期望与方差的关系,既很好地解释了正态分布密度函数的几何意义与参数的概率意义,也适应中学生的认知能力.
In this paper, several important concepts in probability, such as hypergeometric distribution, conditional probability, discrete random variables and distribution column, independent random events and binomial distribution, mathematical expectation and variance, and normal distribution, were created to guide classroom teaching in real problem situations, which provided practical teaching scheme for teachers. In particular, the normal distribution density function was not the same as the formulaic definition in college textbooks and the limit definition of frequency histogram in middle school textbooks. According to the authors, the definition of formulaic was abstract for middle school students, but using the limit of frequency histogram to define normal density function was beyond the knowledge point and cognitive ability of middle school students, and the practical teaching was not operable. This paper starts from the function simulation of the real image, through the similarity between the real graph and the frequency histogram to find out the function to simulate such graph. Then discuss the relation between the parameters and the image, finally, return to the probability, discuss the relation between the parameters and the mathematical expectation and the variance.
In this paper, several important concepts in probability, such as hypergeometric distribution, conditional probability, discrete random variables and distribution column, independent random events and binomial distribution, mathematical expectation and variance, and normal distribution, were created to guide classroom teaching in real problem situations, which provided practical teaching scheme for teachers. In particular, the normal distribution density function was not the same as the formulaic definition in college textbooks and the limit definition of frequency histogram in middle school textbooks. According to the authors, the definition of formulaic was abstract for middle school students, but using the limit of frequency histogram to define normal density function was beyond the knowledge point and cognitive ability of middle school students, and the practical teaching was not operable. This paper starts from the function simulation of the real image, through the similarity between the real graph and the frequency histogram to find out the function to simulate such graph. Then discuss the relation between the parameters and the image, finally, return to the probability, discuss the relation between the parameters and the mathematical expectation and the variance.
引文
[1]苏淳.概率论[M].北京:科学出版社,2010:43-106.
[2]盛骤,谢式千.概率论与数理统计及其应用[M].北京:高等教育出版社,2004:25-68.
[2]盛骤,谢式千.概率论与数理统计及其应用[M].北京:高等教育出版社,2004:25-68.