3. Excavate historical materials and let students experience the thinking method of probability and statistics.
Probability statistics is an important part of the new mathematics curriculum in middle schools. It studies the statistical regularity of random phenomena and has unique concepts, methods and theories. In teaching, we should pay more attention to the process of experiment and statistics, and combine historical examples to cultivate students' random thinking and statistical concepts as soon as possible.
3. 1 distractions
The core of random thought is to understand the statistical regularity hidden behind random phenomena, and emphasize the relationship between the randomness of individual observation of random phenomena and the statistical regularity of a large number of observations. Necessity is always hidden behind contingency, and a large number of random phenomena reflect the inevitability in the development of things. It is through the study of this contingency that random thought finds the inevitability behind it, that is, statistical regularity, and understands and grasps random phenomena through this inevitability.
Random experiment is an important method in random thinking. In order to study the statistical laws of random phenomena, famous random experiments have been carried out in history, such as Buffon and Pearson's coin toss experiment and Gao Erdun's Galton board test model. For example, if we throw a lot of coins, the frequency of facing up is very close to half, which means that the theoretical probability of facing up is 12. We call this phenomenon that individual results are uncertain, but after repeated many times, the results are regular. "Random" is not synonymous with "chance", but describes a sequence different from certainty, while probability statistics is a mathematics that describes randomness and statistical regularity.
The key to understanding random thought is to understand that the test frequency of an event deviates from the theoretical probability, and the existence of deviation is normal. Although the frequency of repeated tests gradually stabilizes to its theoretical probability, it does not rule out that no matter how many tests are made, the test probability is still an approximation of the theoretical probability and cannot be equal to the theoretical probability. For example, in theory, the probability of "randomly throwing a coin and landing face up" is 12. However, 100 tests cannot guarantee that 50 times face up and 50 times face down. As long as students really do experiments, they will certainly realize this. In fact, 100 coin toss test will be exactly 50 times face up, and the probability of 50 times face down is only c? 50? 100? ( 12)? 100? ≈ 8%, far lower than the probability of coin-operated coin facing up once, that is, 50%. In teaching, students should be prevented from intuitively understanding probability as "ratio", so as to have a deeper understanding of the probability of an event.
Random thoughts also include the randomness of sampling in statistical experiments and the randomness of simulation experiments or random sampling results. Only when students realize this, can they truly understand the randomness widely existing in the real world and actively apply it to their lives. There are many sampling methods, but no matter which method is used for sampling, we must adhere to the principle of random sampling. This is the basic requirement to avoid human influence and ensure the objectivity and truth of the samples.
3.2 Statistical inference thought
The core goal of statistics course is to guide students to understand the characteristics and functions of statistical thinking and the difference between statistical thinking and deterministic thinking. For example, in the study of estimating the population by using samples, students should realize that the information provided by samples reflects the relevant characteristics of the population to a certain extent, but there is a certain deviation from the population through the analysis of specific data. On the other hand, if the sampling method is reasonable, for example, Laplace, a famous mathematician, studied the birth laws of boys and girls in London, Petersburg, Berlin and France, and the statistical data obtained showed that the birth frequency of boys fluctuated around 2243 during 10 years; The data of gender composition of the total population in previous censuses in China are very close to those obtained by Laplace.
Scientists have found that not only in human social life, but also in nature, the reproduction and evolution of life obey the law of probability and statistics. As early as 1843, the Czech monk Mendel revealed the mystery of nature to the world for the first time by studying the genetic law of peas. Because the two genes of pea are separated from each other, they do not interfere with each other when entering the next generation of hybrid cells, and finally they are randomly combined in the process of biological pollination. Therefore, this law is also called "separation phenomenon". Later, after arduous exploration, Mendel found that when two pairs of plants with different traits were crossed, the genes of different pairs were freely combined and the opportunities were equal. This is Mendel's second law, also known as the law of free combination. The law of separation and free combination discovered by Mendel is essentially the embodiment of the law of probability and statistics in the genetic process.
The process of statistical reasoning is different from logical reasoning in mathematics. It is a probabilistic reasoning method, and its principle is "small probability events". The principle of small probability events holds that in an experiment, small probability events will hardly happen. For example, the solution of hypothesis testing problem is the embodiment of statistical inference. For a hypothesis, given a small probability level standard, if the sampling data are sorted out and calculated, if the result makes a small probability event happen (which is different from the small probability event), otherwise, the original hypothesis is considered acceptable. The implementation of this statistical inference idea fully demonstrates the practicability of mathematical statistics. In teaching, we can use examples such as drug efficacy test to introduce the idea of statistical inference.
4. Use historical examples of probability model to stimulate students' innovative consciousness.
A large part of stochastic mathematics can be described by probability models, such as finite equal probability model (classical probability model), Bernoulli probability model, normal distribution and so on. The application of probability model method is to simulate and construct a realistic prototype or abstract model according to the specific characteristics of a random problem to reflect the inherent law of the problem, and then choose the corresponding mathematical method to answer the obtained mathematical model. It shows the process from practice to theory and back to practice. In the teaching of probability statistics, we should attach importance to the understanding and application of probability models, downplay complex calculations, let students experience the process of summarizing specific probability models from multiple examples, experience the similarities and differences of these examples, and cultivate students' ability to identify models. David S. Moore, a professor of statistics at Purdue University in the United States, once said: "Learning combinatorics can not enhance our understanding of the concept of opportunity. The ability to develop and use probabilistic modeling is not better than other disciplines. In most cases, we should avoid combination problems unless it is the simplest counting problem. " Using probability model to solve problems is a typical inductive thinking mode, which cannot be separated from people's observation, experiment and reasonable reasoning. It is the embodiment of mathematical consciousness and thinking method, which is helpful to cultivate students' ability to solve practical problems and innovative consciousness by applying mathematical theory.
While the history of mathematics shows the development process of random mathematics knowledge, mathematicians' application of mathematical methods and innovative thinking in solving practical problems often bring inspiration to future generations. For example, finding π with probability model is a typical historical example, and a history of calculating pi is regarded as a "symbol of civilization" of human beings. 1872, the British scholar William Shanks has calculated the value of π to 707 decimal places. After more than half a century, the mathematician Fagerson has doubts about the calculation result of X. Fagerson's doubts are based on the following peculiar ideas: there is no preference for one or two numbers in the value of π, that is, the probability of each number should be equal to 1 10. With the appearance and application of electronic computers, the calculation of π has made rapid progress. 1973, French scholar Jean Gaye. This paper makes an interesting statistic on the frequency of each bit in the first millionth of π, and draws the conclusion that although there are some ups and downs in the appearance of each bit, it is basically equally divided. It seems that Ferguson's idea should be correct, and in the numerical expansion of π, there are: p (0) = p (1) = p (2) = … = p (9) =? 0. 1? But sometimes, because the probability model contains uncertain random factors, it is more difficult to analyze than the deterministic model. In this case, Monte Carlo method can be considered. Monte Carlo method is the basis of computer simulation, and its name comes from the world-famous casino-Monte Carlo in Monaco. Its history originated from a method of calculating pi proposed by French scientist Buffon in 1777, that is, the famous Monte Carlo method of Buffon's needle problem, which belongs to a branch of experimental mathematics. The basic idea is to establish a probability model first, so that the solution of the problem happens to be the parameters or other related characteristics of the model. Then, the percentage of an event is counted through simulated statistical experiments, that is, multiple random sampling experiments. As long as there are many experiments, the percentage is similar to the probability of an event. Finally, the parameters to be estimated are obtained by using the established probability model, that is, the solution of the problem.
refer to
1 Li Wenlin. Introduction to the history of mathematics [M]. Beijing: Higher Education Press, 2002.
Zhang Dan. Statistics and probability [M]. Beijing: Higher Education Press, 2006.
Three Zhang Yuannan. The story of probability and equation [M]. Beijing: China Children Publishing House, 2005.