Middle School Students' Misunderstandings Of Probability?the Perspective Of HPM

Probability theory which studies the quantitative law of random phenomena is an important branch of mathematics.The probability plays an more and more important role in the new curriculum.Moreover,the students' ability of data analysis and logical reasoning in probabilistic learning are the key research contents of the Core Literacy.Probabilistic teaching is the focus of mathematics curriculum reform in our country in recent years.The mistakes in probabilistic teaching or students' learning have always been the focus of teaching research.Therefore,it is necessary to investigate and analyze the main errors in probability understanding of junior high school students.In order to explore students' errors in probabilistic learning and the response measures,this paper puts forward the following three research questions:(1)What are the major errors of grade 9-11 students in the probabilistic learning ?And what are the causes of the errors?(2)What are the differences in the distribution of error types between grades and genders on probabilistic issues?(3)Can the probabilistic teaching from the perspective of HPM based on students' errors effectively reduce students' errors ?This study is divided into four parts: data collection,coding analysis,teaching practice and putting suggestions.This research is divided into two parts: investigation and teaching practice.Based on the HPM theory,a total of 822 questionnaires including pre-study,main research and post-test questionnaire were conducted among497 students in Grade 11~12 and 291 students in Grade 9 and Grade 12 in Shanghai,Zhejiang Province.By coding and analyzing the questionnaire data using spss19.0,we found that there are 11 kinds of errors in students' understanding of probability: Equal Possibility Bias 1,Equal Possibility Bias 2,Uncertainty Statement,Subjective Judgment,Frequency is Probability,Proportional Reasoning Error,Wrong Prophecy,Simple Compound Error,Probability 0 Doesn't Occur,Representational Heuristic,Conforming the Trend or Bucking the Trend.The most common errors areas follows: Equal Possibility Bias 1 & 2,Uncertainty Statement,Simple Compound Error and Representational Heuristic error.Due to the weakening probability value in senior high school,the example models always easily make students have the wrong understanding that "the event has only two sides".The reason why there is more Equal Possibility Bias in senior grade 2 is that the probability model of high school students' contact is limited to the classical probability,the events are listed,the probability can be calculated by counting the number,and the possibility is not necessary to be considered."Uncertainty Statement" is the result of students' lack of understanding of the value of probability.Students with "Simple Compound Errors" are prone to regard the stepwise test as a step test and jump to conclusions because of the trouble of the proportion value.Representational Heuristic is a misconception arising from the deficiency of life experience,and the students who hold this view think that the more general or special it is,the more likely it is to happen.Gender had less influence on the distribution of probability errors.The ninth grade students have a high level of understanding and calculation of the simple probability model,at the same time,the probability learning of the senior students has deepened the their understanding of the law of large numbers and independent events,improved their understanding and computing ability of the complex probabilistic model.The high school probability teaching pays attention to the extension yet,doesn't cause the student's appreciation of the concept.Teaching practice is based on the results of investigation and research,combined with HPM theory and inquiry teaching,trying to integrate mistakes into teaching.The teaching feedback of "Classical Models of Probability" shows that there are fewer errors of Equal Possibility Bias and Representational Heuristic,and students can distinguish the difference of basic event space by using the method of arrangement and combination.Three enlightenments about teaching can be obtained from the effect of teaching practice:(1)Pay attention to students' errors in probabilistic learning,especially the "Equal Possibility Bias" error.There are many profound probabilistic questions in the history of probability development,among which there are mathematicians' errors,which can be copied or adapted to help students understand their errors;(2)Pay attention to students' understanding of concepts,and the connection of probability knowledge in middle and high schools,and calculating probability by exhaustive method is the basis.The probability experiment in the classroom is necessary,and the actual hands-on operation records the data,let the student experience more personally the relation between frequency and probability under the few experiments and the large number of experiments;(3)Make students understand the history of the development of probability,it is a spur to make students know that the knowledge of probability is the result of the participation of human in the cultural activities,and the students are also involved in the progress of mathematics,which is an effective means to carry out the humanities teaching of probability humanities.The enlightenment of textbook compilation is as follows: the probability model should be added to the textbook,the depth of probabilistic teaching should be improved,and the requirements of the content of probability should be raised appropriately in the examination;The arrangement of probability contents in Shanghai Education Edition conforms to the order of history and accords with the students' cognitive foundation,but the reading materials should be used to stimulate the students' interest in learning more effectively and to give full play to the educational value of the history of mathematics.