The history of mathematics is a tool for learning and understanding mathematics. If people want to understand the development process of mathematical concepts, ideas and methods and establish the overall consciousness of mathematics, they must take the history of mathematics as a guide. Probability theory and mathematical statistics also have their own history of continuous development and perfection. At present, China is promoting the reform of basic education, and attaches great importance to the education of mathematics history and mathematics culture. Applying the history of mathematics in the teaching of probability and statistics in middle schools will help students understand the relationship between mathematical knowledge and the unique thinking methods of uncertain mathematics, thus improving their ability.
1 Interpret historical facts and promote students' understanding of the definition of probability.
The classical definition of probability is given by Laplace in 18 12, and its discussion object is limited to the case that all possible results in random experiments are finite and equal. In teaching, we can combine the problem of "gambling distribution" to experience the model characteristics of classical probability and deepen our understanding of the definition. Give a simple case of this problem: A and B gamble, each betting on 30 yuan and 60 yuan. Both are 12. Everyone agrees that whoever wins three games first will win all the bets. Now, 60 yuan has gambled for three games, and A 2 won 1, but he stopped gambling for some reason. How should this 60-dollar bet be divided between two people is fair. At first glance, he thought it should be distributed according to 2: 1, that is, A gets 40 yuan and B gets 20 yuan. Some people have put forward some other solutions. The correct division should take into account the probability that A and B will continue to gamble on this basis and finally win. In fact, you can decide the outcome by playing two more games at most, and these two games have four possible outcomes: A, B, B, and B. The first three situations are that A wins in the end, and only B wins, with a ratio of 3∶ 1, so the fair distribution of bets should be 3.
The classical definition of probability has the advantage of computability, but it also has obvious limitations. It needs limited sample points. If there are infinite sample points in the sample space, the classical definition of probability is not applicable. When the finite sample points are extended to infinite sample points, geometric probability is introduced. In this way, a geometric method for determining probability is formed. The most typical examples of the geometric definition of learning probability are the "encounter problem" and the famous "Buffon throwing needle experiment" in history: draw some parallel lines on a plane, and the distance between them is equal to a, and throw a needle with a length of L(L is less than A) into this plane at will. Try to find the probability that this needle intersects any parallel line. This geometric probability problem can be solved by integral operation. Because the theoretical probability of Buffon's needle-throwing experiment contains the constant π, we can design L and A in teaching, estimate the probability P through statistical experiments, and then use the probability model formula given above to find pi. In this way, the geometric definition of probability is organically linked with the learning of statistical definition of probability, and at the same time, students can experience the relationship between the diversity of methods for finding π and mathematical knowledge.
Both the classical definition and geometric definition of probability require that the probability of basic events in random experiments is equal, but it is found that the ratio of the number of times n of an event to the total number of experiments will be stable near a constant when the number of times n of experiments is large under the same conditions. The greater n is, the less likely it is that this ratio will "get away" from this constant. This constant is called the probability of this event. This definition is closely related to statistics and is based on the stability of frequency, so it is called the frequency definition of probability. The object of this probability discussion is no longer limited to all random experiments with equal possible results, so it is more general. Based on the statistical experiment of students throwing coins by hand and referring to the results of many times of throwing coins by famous scientists in history, we can further feel the requirements of large-scale experiment of frequency probability, as well as the randomness and statistical regularity of probability statistics.
It is easy to see from the following table that the frequency fluctuates greatly when the number of throws is small, and it is stable when the number of throws is large, that is, the frequency of the frontal face swings around 0.5 and gradually stabilizes at 0.5. These three definitions of probability are descriptive, and the word "possibility" is used in the narrative. Probability is only about the concept of "possibility", so these definitions are not strict in theory. Due to the lack of strict theoretical basis, people often find some loopholes to drill, the most typical of which is the probability paradox put forward by French mathematician bertram in 1889: What is the probability that the length of any chord in a circle exceeds the side length A of an equilateral triangle inscribed in the circle? The author gives three different answers:
The first solution is that when the chord midpoint H is evenly distributed on the diameter PQ, P= 12 (Figure1);
The second solution of Figure 1 Figure 2 and Figure 3 is that when the chord midpoint H is evenly distributed on a small circumference, P= 13 (Figure 2);
The third solution assumes that the midpoint H of the chord is evenly distributed in a small circle, P= 14 (Figure 3). The fundamental reason for this paradox is that the equipotential assumptions of the three solutions are different and the corresponding sample spaces are also different. They are three different random experiments. Therefore, in the case of infinite sample points, sample space and sample points must be specifically defined, and the axiomatic definition of probability came into being. teaching
In recent years, with the development of mathematics, people pay more and more attention to subjective probability. The subjective definition of probability is also called intuitive definition. "It refers to the cognitive subject's quantitative judgment on the possibility of a certain situation according to his own knowledge, information and evidence" (Chen Xiru, 2000,6). The "Bayesian formula" put forward by British scholar Bayes is considered to be the first formula to use subjective probability. The problem is that in practice, because the process under consideration has not been carried out, it is often impossible to get the probability of many things. But in fact, if people analyze the obtained data according to previous empirical data or even subjective or objective requirements, estimate an optimal value as the hypothetical probability of the research population, and finally correct the hypothetical probability on the basis of obtaining new information, this is beyond reproach. In the increasingly complex economic activities in modern times, when some decisions cannot be judged by theoretical probability or empirical probability, it is feasible to apply subjective probability to economic decision-making problems such as investment. Appropriate introduction of subjective probability in teaching can enrich students' understanding of probability.
2. Analyze historical situations and cultivate students' correct probability intuition.
British scholar Wells said: "Statistical thinking method, like reading and writing ability, will one day become an essential ability for efficient citizens." However, probability statistics is different from the branch of mathematics that studies deterministic phenomena such as geometry and algebra, and has its own unique style in theory and method. In the study of probability and statistics, students will encounter many random mathematical theories. Because all kinds of random phenomena can't be strictly controlled and accurately predicted by "causality" and can't be summarized by some simple laws, they should be comprehensively analyzed from a large number of observations to find out the regularity, so it is necessary to cultivate students' correct thinking methods of probability and statistics.
In teaching, we often find that many students are often confined to the thinking mode of deterministic mathematics, unable to establish correct probability intuition, and there are a lot of misunderstandings in probability learning and problem solving. In fact, for teachers, it is an important issue that must be handled well to maintain the logical rigor of probability and statistics courses and pay attention to the cultivation of students' probability intuition ability. Let students experience the close connection between probability and actual things as soon as possible. A keen sense of randomness in real things is a necessary condition for establishing correct probability intuition. For example, when studying "Birthday Question", the teacher can introduce the following historical information first: There are 42 presidents in American history so far, including Polk of 1 1, Harding of the 29th, all with birthdays of 1 1, and Adams.
The "birthday problem" can be very confusing: two out of 50 people have the same birthday, which you may think is just a coincidence. In fact, it is almost certain that at least two people have birthdays on the same day. We can use the probability method to calculate. For the sake of simplicity, we don't remember leap years, and a year is 365 days. So the theoretical probability of this problem is 1-A50-A50? 365? 36550? ≈ 0.97. The probability of this happening is not as small as most people intuitively think, but quite large. This example tells us that the usual "intuition" is not very reliable, which strongly illustrates the importance of studying the statistical laws of random phenomena. The false intuition in this example comes from people's subconscious intuition that two out of 50 people have the same birthday and some out of 50 people have the same birthday. The theoretical probability of the latter case is only 65,438. ≈ 0. 13. So the probability that "two out of 50 people have the same birthday" is not a big illusion. In teaching, students can go through the process of estimating and verifying the probability of random events through statistical investigation or random simulation experiments, and gradually establish correct probability intuition.
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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 test does not guarantee 50 times of alignment and 50 times of alignment. As long as students really do the test, they will certainly realize this. In fact, the probability of 100 coin toss test being right 50 times and wrong 50 times is only C50? 100? ( 12) 100? ≈ ? 8%, far lower than the probability of coin-operated coin facing up last time, 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 of the model or other related characteristics. 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.
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