A Study On The Development Of A Learning Progression For Probability Concepts Of Middle School Students

In the current era,probabilistic literacy has become an indispensable mathematical literacy for every member of society.Therefore,in order to cultivate probabilistic thinking,probability content is regarded as one of the core concepts of mathematics and runs through the whole stage of basic education.However,whether it is in largescale evaluation projects such as TIMSS and PISA,or in basic education quality monitoring in China,it is found that compared with “number and algebra”,“graphics and geometry” and other parts,students perform poorly in the part of “probability and statistics”.In addition,previous researches mostly focused on the investigation of single knowledge point,and paid little attention to the internal structure of probability.Therefore,the researches on the cognitive structure of students' probability concepts are relatively weak.Nowadays,the construction of learning progression for core concepts is an important trend in the development of current international education.In order to keep up with the dynamic trend of international education research on students' learning and evaluation,this study attempts to establish a model for students' cognitive development in probability,so as to more truly reflect the development process of students' thinking on the concepts of probability.In view of this,this study takes students from grades 7 to 11 as the research objects,and the problem solving of the“probability concepts” as the research topic.On this basis,this study tries to construct a hypothesized learning progression for probability concepts based on Cognitive Diagnosis Theory,and then develops a formal learning progression for middle school students.Finally,it is applied to the diagnosis and evaluation of students' probabilistic thinking to describe their learning performance in detail,so as to promote the integration of curriculum,teaching and evaluation.Research question 1: How to develop a hypothesized learning progression for probability concepts based on Cognitive Diagnosis Theory?The main research methods of this issue are literature review and expert interview.Firstly,the attributes(knowledge,skills,strategies,etc.)needed in the process of solving the problem of “probability concepts” are extracted.Five basic concepts are identified: randomness,sample space,probability comparisons,probability of an event,and probability estimation,and nine cognitive attributes are extracted from them: A1-Randomness,A2-One-dimensional sample space,A3-Two-dimensional sample space,A4-One-dimensional probability comparisons,A5-Two-dimensional probability comparisons,A6-Probability of a one-dimensional event,A7-Probability of a twodimensional event,A8-Probability estimation of a one-dimensional event,A9-Probability estimation of a two-dimensional event.Secondly,the attribute hierarchy between the extracted attributes is established.Finally,based on this,the progression dimensions,the progression levels and the expected students' learning performance of the hypothesized learning progression are determined,and the hypothesized learning progression for probability concepts is formed.Research question 2: How to verify and modify the hypothesized learning progression for probability concepts based on the G-DINA model?The main research methods of this issue are literature review,expert interview and test.Firstly,the -matrix is determined and a cognitive diagnostic test of probability concepts is developed,which consisted of 26 items.All items are dichotomous,with the correct score of ‘1' and the wrong score of ‘0'.And the time allocated to the test is40 minutes.Secondly,the rationality of the extracted attributes,the established attribute hierarchy and the hypothesized learning progression is verified according to various data analysis results.After testing,the extracted attributes,the established attribute hierarchy are basically scientific and reasonable.The cognitive diagnostic test of probability concepts(revised version)meets psychometrics standards.The setting of the hypothesized learning progression is basically reasonable,in which the students' performance in A8-Probability estimation of a one-dimensional event is lower than expected.According to the attribute mastery probability,it is adjusted from Level 2 to Level 3 in the learning progression.The formal learning progression for probability concepts is finally developed,which is used to evaluate the learning performance of middle school students.Research question 3: What is the learning performance of middle school students by using the learning progression for probability concepts?The main research method of this issue is test method.Firstly,this paper analyzes the learning progression levels of probability concepts for middle school students.The results show that students' understanding of probability concepts is constantly developing and improving,and their understanding of one-dimensional probability concepts has developed rapidly,while their understanding of two-dimensional probability concepts has developed relatively slowly.The academic performance of the students in grade 8 is lower than that of grade 7,but there is no statistical difference.Secondly,the cognitive structure of middle school students' probability concepts is analyzed.The results show that the Attribute Master Patterns of middle school students' probabilistic thinking accumulates towards the terminal point.To be specific,with the rise of grade,the main Attribute Master Patterns of students are decreasing and becoming more and more concentrated,from the initial 10 or so to 4.At the same time,the number of attributes mastered by students is gradually increasing.From 3 to 6 in grades 7 and 8,until grade 11,students have basically mastered 8 or 9 attributes.And the proportion of students who reached the terminal point has also been a substantial improvement.In addition,most of the cognitive disadvantages of middle school students in the probability concepts can be transformed into cognitive advantages.Finally,this study develops the design of multiple learning pathways for the probability concepts of middle school students,and makes a case analysis according to the main Attribute Master Patterns and individual cognitive diagnosis of students respectively,providing a variety of learning pathways.In conclusion,the innovation of this study is as follows: Cognitive Diagnosis Theory is introduced into the construction process of learning progression for probability concepts,and the formal learning progression is applied to the evaluation of students' learning performance,so as to develop personalized remedial plans for students.The final results also confirm the feasibility and superiority of using cognitive diagnosis models to develop the learning progression.At the same time,it is not difficult to find that the combination of learning progressions with Cognitive Diagnosis Theory has both great advantages and certain difficulties.On the one hand,this research provides experience for the development of a learning progression based on Cognitive Diagnosis Theory in future.On the other hand,the learning progression developed by this research can provide rich cognitive diagnostic information for the evaluation of students' performance,which is helpful for the transformation of learning progression results to teaching practice,and can also provide possibility for students' selfimprovement.However,there are still great challenges in this process,and it needs the support of multiple experts and further exploration.