Junior High School Students’ Understanding Of Irrational Numbers

A lot of empirical researches show that there are many problems on middle school students’ and pre-service teachers’understanding of irrational numbers. The first class of irrational numbers appears in junior high school, students’understanding of irrational numbers is the major question in this paper. In order to avoid repeating previous’work, this paper based on the dual nature of mathematical concepts, adding a historical perspective on students’ understanding of irrational numbers.This paper analyzed the history of irrational numbers, as well as definitions of irrational numbers in 30 early American textbooks. On the basis of literature research and pre-tested on mathematical teachers, a set of test questions for students’understanding of irrational numbers were developed. A questionnaire and interview were conducted among 198 students of eight and nine grade in a junior high school of Shanghai. And then the test results were analyzed from tree aspects:concept definition and concept image of irrational numbers, students’ understanding of irrational numbers, and dual nature of mathematical concepts. We reached the following conclusions:(1) Most students know exactly the definition of irrational numbers, the proportion of eighth-grade students who master the definition of irrational numbers is higher than ninth grade. "Irrational number is infinite non-repeating decimals" is highly recognized by junior high school students and teachers. Students’concept definition and concept image on irrational numbers are often related to the number line, root, approximate values and so on. Students’ concept image on irrational numbers is not static, but change over time.(2) When judged against irrational numbers related properties, students’performance is not ideal. Most of the students is overly dependent on the decimal form of irrational numbers to determine if a number is an irrational number. Students did not attract enough attention to incommensurability, they are not aware of the importance of incommensurability in the development of irrational numbers. Some students are lack of recognition on the incommensurability in real life.(3) Whether in the historical development of the irrational number, or the students’ understanding of the irrational number reflects the dual nature. Obstacles on the understanding of irrational numbers occur easily if the students stay too long on predominantly operational approach. When contradiction of knowledge and belief at this stage has not been resolved, much problems of the understanding on irrational numbers would appear.Finally, some suggestions for teaching are made based on the above conclusions, providing some reference value for mathematical teachers on the teaching of irrational numbers.