Introducing the history of mathematics into the classroom, widely applying cases in teaching, actively carrying out random experiments and guiding students to actively explore are helpful to improve the teaching methods of probability theory, solve practical teaching problems and improve teaching quality. Diversified teaching methods and rich teaching contents can deepen students' knowledge and understanding of objective random phenomena and stimulate students' spirit of independent learning and active exploration.
There are three major leaps in the historical development of mathematics. The first leap is from arithmetic to algebra, the second leap is from constant mathematics to variable mathematics, and the third leap is from deterministic mathematics to stochastic mathematics. The randomness of the real world makes it natural for all fields to change from deterministic theory to stochastic theory. Moreover, the tools, conclusions and methods of stochastic mathematics have opened up a new way to solve the problems in deterministic mathematics. Therefore, it can be said that stochastic mathematics will surely become one of the highlights of mainstream mathematics in the future. Probability theory, as the most basic part of stochastic mathematics, has become a compulsory basic course for many college students. However, one of the main problems in the teaching process is that students are often used to the thinking mode of deterministic mathematics. It is considered that the basic concepts in probability are abstract and difficult to understand, and the limitations of thinking are difficult to develop. All these make students flinch from this course, so it is very important to cultivate students' thinking method of learning random mathematics in the teaching process of probability theory. The purpose of this paper is to introduce our reform attempt in this course in order to attract more attention.
1 Integrating the history of mathematics into the teaching classroom In the process of probability theory teaching, introducing relevant history of mathematics can help students better understand that probability theory is not only a "Chun Xue", but also a subject with strong application background. For example, the most important distribution in probability theory-normal distribution was in18th century. It is put forward to solve the astronomical observation error. In the 17 and 18 centuries, astronomical observation error was an important problem, and many scientists studied it. The concept of normal distribution was put forward by German mathematician and astronomer democritus in 65438+. German mathematician Gauss took the lead in applying normal distribution to astronomical research, pointing out that normal distribution can "fit" error distribution well, so it is also called Gaussian distribution. Today, normal distribution is the most important probability distribution and the most widely used continuous distribution. 1844 during the French conscription, many people who met the age of enlistment claimed that their height was lower than the minimum height requirement for conscription. So you can be exempted from military service. Someone must have lied to avoid military service. Sure enough, the Belgian mathematician kettler (A. Quetlet, 1796- 1874) compared the height distribution of candidates and ordinary men by using the law that height obeys normal distribution. We found 2000 people in France who pretended to be below the minimum height requirement in order to avoid conscription [1]. In the university stage, we not only hope to stimulate students' interest in learning probability theory by presenting the history of mathematics in the teaching classroom, but also pay more attention to let students dig deep into the history of mathematics through interest and feel the thinking method of random mathematics [2]. We know that classical probability theory needs limited sample space. Geometric probability can eliminate this condition, which is easy for students to understand. But students always think that the axiomatic definition of probability is abstract and unacceptable, especially σ algebra [3].
This concept: Let ω be a sample space. What if some subsets of ω are a set? Meet the following conditions: (1) ω ∈? ; (2) if A∈? , and then a ∈? ; (3) if ∈ n A? , n = 1, 2, then ∑∞= nnA∪ 1? , then what shall we say? Is σ algebra of ω. In order to make students understand this concept better, we can introduce a little history of geometric probability to introduce why we should establish an axiomatic definition of probability and why we need σ algebra. Geometric probability is a new probability calculation method developed at the end of 19, which is a further development on the basis of classical probability and an extension of the concept of equal possible events from finite to infinite 35438+0899. French scholar Bertrand put forward the so-called "Bertrand paradox" [3]. For the concept of geometric probability itself, this paradox is: given a radius of 1
What is the probability that the chord length is not less than 3? For this problem, if it is assumed that the endpoints are evenly distributed on the circumference, the probability is equal to1/3; If the diameter of the midpoint of the chord is assumed to be evenly distributed, the probability is1/2; If the midpoint of the chord is assumed to be evenly distributed in a circle, the probability is equal to 1/4. There will be three different answers to the same question, because different equal possibility assumptions are adopted when choosing chords! These three answers are for three different random experiments, and they are all correct for their own random experiments. Therefore, when using terms such as "random", "equal possibility" and "uniform distribution", their meanings should be clearly expressed, which varies from experiment to experiment. That is to say, when we assume that the endpoints are uniformly distributed on the circumference, we cannot consider the events corresponding to the uniform distribution of the midpoint of a chord in diameter or the uniform distribution of the midpoint of a chord on a circle. In other words, when we assume that the endpoints are evenly distributed on the circumference, we only regard the elements corresponding to the evenly distributed endpoints on the circumference as events. Now let's understand the concept of σ-algebra: for the same sample space ω,? 1 ={? , ω} is its σ algebra; Let a be a subset of ω, then? 2 ={? , a, a, ω} is also σ algebra of ω; Let b be another subset of ω different from A, then? 3 = {? , a, b, a, b, ab, ab, ba, ab, ω} is also σ algebra of ω; The set of all subsets of ω can also form σ algebra of ω. When we think about it. 2. Just put elements? The element of 2? , a, a, ω as events, but b or AB is not considered. Therefore, the definition of σ algebra is easier to understand. The case teaching method is widely used, which is different from the general examples and has the actual background that causes problems and can be understood by students. Case teaching method is to use cases as teaching tools to guide students to solve practical problems. Through analysis and discussion, this paper puts forward the basic methods and ways to solve the problem, a teaching method. The basic knowledge of probability theory can be introduced from an intuitive, interesting and easy-to-understand perspective. Speaking of conditional probability, we can first introduce an interesting case-"Marilyn Question": "Marilyn was lucky to answer first" in the United States more than ten years ago.
The radio station announced such a topic: two sheep and a car are hidden behind three doors (for example, 1, No.2, No.3). If you guess the door of the hidden car correctly, the car is yours. Now let you choose, for example, you choose 1, and then the host opens one of the other two doors to let you see clearly that there is a sheep behind this door.
Because this question is very similar to some entertainment quiz programs on TV, students actively participate in the discussion of this question. The result of the discussion is that the answer to this question is related to whether the host knows everything behind the door, so conditional probability can naturally be introduced. Learning new concepts in such a warm atmosphere, on the one hand, makes students' enthusiasm soar. On the other hand, making students realize that the knowledge of probability theory is closely related to our daily life can help us solve many practical problems. Therefore, when introducing the basic knowledge of probability theory, introducing classic cases, such as gambling books, inventory and income, privacy investigation, probability and password,17th century Central American witchcraft, sensitive investigation, blood test, 65433, etc., will achieve good results.
Probability theory can not only provide solutions to the above problems, but also give theoretical explanations to some random phenomena. Because of this, probability theory has become an effective tool for us to understand the objective world. For example, we know that it is extremely unlikely that a particular person will become a great man. One reason is that someone's birth is a combination of a series of random events: the combination of parents, grandparents, grandparents ... and the meeting of two heterosexual germ cells, which must contain some factors that produce genius. Another reason is that after the baby is born, all kinds of accidental experiences must be conducive to his success as a whole. Everyone should provide him with a good opportunity for his time, his education, his activities, the people and things he contacts. Even so, great men have come forth in large numbers. Although the probability of a person's success is extremely small, there are always outstanding people among billions of people. This is the so-called "one meaning". How to explain this problem with the knowledge of probability theory? Let the probability of event A in an experiment be ε, 0.
(1) can be repeated under the same conditions; (2) There is more than one possible result of each test, and all possible results of the test can be specified in advance; (3) Before the experiment, we can't be sure what kind of results will appear. When teaching the definition of random experiment, we often list the above three characteristics one by one, and then give a few simple examples to illustrate the end. But after watching a foreign popular science short film, we were very inspired. The content of the program is to verify: when bread with butter on one side and nothing on the other side falls from the table, which side of it faces up? To our surprise, in order to make the test results more convincing, the experimenter specially made a machine for buttering bread and a bread slinger, and then began to do experiments. No matter what the conclusion of this question is, what we observed is that they designed the experiment quite rigorously to ensure that the random experiment was repeated under the same conditions. We introduce this popular science short film into classroom teaching and analyze it with examples. Three characteristics of random experiment are put forward. Students accept it naturally and the whole teaching process becomes relaxed and happy. Therefore, simple tools can be used in teaching to make the theoretical knowledge as intuitive as possible, such as the application demonstration of total probability formula, the graphic representation of geometric probability, the distribution of random variable function, the statistical significance of mathematical expectation, two-dimensional normal distribution, Galton nail plate experiment and so on. And give an abstract theory in an intuitive form. Deepen students' understanding of theory. But it is impossible for us to realize every random experiment in the limited classroom time. Therefore, in order to effectively stimulate students' thinking in images, we have adopted the means of multimedia-assisted theoretical teaching, and established a vivid and intuitive teaching environment with illustrations, audio-visual combination and combination of numbers and shapes through computer graphics display, animation simulation, numerical calculation and text description, which has broadened students' thinking. It is helpful to master the basic theory of probability theory. At the same time, let students experience the charm of modern teaching in the process of accepting theoretical knowledge, and achieve the teaching effect that traditional teaching can not achieve [6]. Guide students to actively explore traditional teaching methods. Teachers often fill their mouths with water in class, and the method is single, only paying attention to the accumulation of students' knowledge. Teachers are the main body of teaching and pay attention to the teaching process. However, it ignores that teaching is an interactive process between teaching and learning. Comparatively speaking, modern teaching methods pay more attention to tapping students' learning potential, aiming at developing students' intelligence to the maximum extent. For example, after giving the definition of conditional probability, we know that when P (a) >; 0, P(B | A) may not be equal to P(B). But once P(B | A) =P(B), it means that the occurrence of event A does not affect the occurrence of event B. Similarly, when p (b) >: 0, if P(A| B) = P(A), it means that the occurrence of event B does not affect the occurrence of event A. Therefore, if p (a) >; 0, P(B)>0, P(B | A) = P(B) and P(A| B) = P(A) all hold, that is to say, the occurrence of these two events does not affect each other. We can ask students to think positively whether they can define the independence of two events as follows:
Definition 1: Let A and B be two random events, if P (a) >; 0, P(B)>0, and we have P(B | A) = P(B) and P(A| B) = P(A), then event A and event B are independent of each other. Next, we can continue to guide students to carefully examine whether the condition p (a) in the definitions of 1 > 0 and P(B)> 0 is an essential requirement. In fact, if p (a) >; 0, P(B)>0, we can get:
P(B | A) = P(B)？ P(AB) = P(A)P(B)？ P (a | b) = p (a)。 But what happens when P(A) = 0 and P(B) = 0? From the relationship between events and the nature of probability, we know AB? a，AB？ B, so P(AB) = 0 = P(A)P(B), the equation still holds. So we can give up the definition of condition P (A) in 1 > 0, and P(B)>0, that is, the definition of event independence is as follows:
Definition 2: Let A and B be two random events. If the equation P(AB) = P(A)P(B) holds, then A and B are independent events, that is to say, A and B are independent of each other. Obviously, definition 2 is more concise than definition 1. In the process of finding this definition, students can not only be encouraged to think positively, but also be well trained and trained in their ability to ask, analyze and solve problems, so as to experience mathematical thoughts and feel the beauty of mathematics. Through practice, we find that introducing the history of mathematics into the classroom can not only make students deeply understand the formation and development process of random mathematics, but also truly feel the thinking method of random mathematics; Applying cases to teaching and conducting random experiments in class can visualize the basic knowledge of probability theory, increase the interest of the course and make it easy for students to understand and master; Guiding students to actively explore can strengthen the interactive process between teaching and learning and stimulate students to use mathematical ideas to solve problems encountered in probability theory. In short, in the teaching of probability theory, we should pay attention to cultivating students' thinking methods of learning stochastic mathematics, and deepen students' understanding and understanding of objective stochastic phenomena through the diversification of teaching methods and the enrichment of teaching content. In addition, we should take talent training as the foundation and realize teacher-centered teaching. The teaching concept of combining subject with object and taking students as the main body will implement the concept of cultivating students' practical ability, innovative consciousness and innovative ability, thus achieving the goal of maximizing students' interests and laying a good foundation for students to engage in the research of economics, finance, management, education, psychology, communication and other disciplines in the future.
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[[ 1] C R R law. Statistics and truth [M]. Beijing Science Press 2004.
[2] Zhu Zhe, Song Naiqing. Integrating the History of Mathematics into Mathematics Curriculum [J]. Journal of Mathematics Education, 2008,17 (4):1–14.
[3] Wang Zikun. Probability theory and its application [M]. Beijing: Beijing Normal University Press, 2007.
[4] Zhang Dianzhou. Random phenomena in the vast universe [M]. Nanning: Guangxi Education Press, 1999.
[5] Wang Zikun. Stochastic Process and Modern Mathematics [M]. Beijing: Beijing Normal University Press, 2006.
Deng Hualing, Fu, Ren Yongtai. Exploration and Practice of Probability Theory and Mathematical Statistics Experiment Course [J]. College Mathematics, 2008,24 (2):114.
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