Hypothetical Learning Trajectory Based On Cognitive Diagnosis Theory

Functions are an important component of the high school mathematics knowledge system.Due to the abstract and obscure nature of function knowledge,teachers often need to use specific function models to assist students in understanding in the classroom.Periodicity is a common property,but students are unable to construct the periodicity of functions with basic elementary functions.Therefore,it is particularly important to introduce trigonometric functions as a typical periodic function.The new curriculum reform proposes that teachers should design appropriate teaching tasks based on students’ cognitive development and thinking level,in order to help students gradually master mathematical knowledge and skills.Based on this,this study attempts to construct a model that can be used to characterize high school students’ cognitive level of trigonometric functions,in order to understand their cognitive process in constructing trigonometric function knowledge.This study takes the hypothetical learning trajectory of constructing trigonometric functions as the research topic,based on cognitive diagnosis theory to construct a hypothetical learning trajectory of trigonometric function knowledge,and takes high school students in a school in Xiamen as the research object to explore their learning trajectory of trigonometric function knowledge.In order to address the question what is the hypothetical learning trajectory for trigonometric functions based on cognitive diagnostic theory,this study uses three research methods: literature analysis,expert interviews,and testing.First,the cognitive attributes of trigonometric functions were extracted through literature analysis.Then,experts were interviewed to confirm the cognitive attributes included in trigonometric functions.The confirmed cognitive attributes were: A1-Arbitrary Angle And Radian Measure,A2-Concept Of Functions,A3-Concept Of Trigonometric Functions,A4-Sum And Difference Formulas,A5-Basic Relationships Of Trigonometric Functions,A6-Graphs And Properties Of Trigonometric Functions,A7-Trigonometric Identity Transformations,A8-Functions,and A9-Applications Of Trigonometric Functions.Next,the hierarchical relationship between the nine cognitive attributes was determined to establish the endpoint of the learning trajectory.Correspondence was established between students’ cognitive levels,learning performance,and cognitive attributes to construct the hypothetical learning trajectory for trigonometric functions.Two sets of trigonometric function knowledge tests were then developed based on the hierarchical relationship between cognitive attributes.One set was used before review,and the other set was used after review,with both sets containing 27 questions.If the respondent answered correctly,one point was awarded;if the respondent answered incorrectly,no points were awarded.The tests were conducted under closed-book conditions,and the time limit was 45 minutes.Based on the data from the small-scale preliminary test,the cognitive attributes and hierarchical relationships determined in this paper were reasonable,and the test questions were appropriate for investigating the learning situation of trigonometric functions among high school students in a certain school in Xiamen.In order to address the question what is the learning trajectory for high school students studying trigonometric functions to test and revise he trigonometric function hypothetical learning trajectory,this study used a test method and found through data analysis that high school students in their second year who did not receive review had a better average mastery rate for the cognitive attribute of the function concept compared to their first year,while their performance on other attributes was worse than the first year.Based on the students’ attribute mastery patterns,they were grouped and a series of problems were designed to guide them in reconstructing their knowledge of trigonometric functions.After the review,the students were tested again and it was found that those who reviewed trigonometric function knowledge had better academic performance than those who had just completed the trigonometric function unit in their first year or those who did not receive a review.Specifically,the reviewed students performed well in five knowledge points: A1-Arbitrary Angle and Radian Measure,A2-Concept of Function,A3-Concept of Trigonometric Function,A4-Induction Formula,and A5-Fundamental Relationships of Trigonometric Functions.Their mastery of A6-Trigonometric Functions and Their Properties and A8-Functions was also good.Although their performance in A7-Trigonometric Identity Transformations and A9-Applications of Trigonometric Functions was not ideal,it was better than their performance in the other two time periods.This also demonstrates that students’ construction of knowledge of trigonometric functions spirals upward rather than rising linearly.It was also found that when students mastered A1-Arbitrary Angle and Radian Measure and A3-Concept of Trigonometric Function,it was more helpful for them to understand the formulas and properties of trigonometric functions and their images.This also indirectly indicates that the concept of trigonometric functions is the most important concept in the trigonometric function unit.Finally,the trigonometric function hypothetical learning trajectory based on cognitive diagnosis theory is tested and revised according to the students’ learning process.In conclusion,the hypothetical learning trajectory for trigonometric functions based on cognitive diagnosis theory can can provide some reference for teachers to assume students’ learning process of trigonometric functions,so as to design teaching activities in line with students’ cognitive rules.In addition,this article also designed test papers that can be used to diagnose students’ trigonometric knowledge,enriching the tools for diagnosing the cognitive development rules of students’ trigonometric functions.