Pre-service And In-service Mathematics Teachers' Understanding Of Irrational Numbers

In this paper, the author takes a survey on pre-service and in-service mathematics teachers'understanding of irrational numbers by using questionnaire and interview. This investigation focuses on the following six aspects such as concept definition and concept image of irrational numbers, the relationship between irrational numbers and real number system and between irrational numbers and real number axis, incommensurability and the comparison of rational and irrational numbers. From the survey the author obtains the following results:a. Although there're several types of definition for irrational numbers, the subjects are limited to the infinite non-repetition decimal type which constrains their understanding and causes them less than ideal performance on the proof of irrational numbers.b. In their concept image, radical type of definition takes first place, then theπ-and e-type, and the logarithmic, triangular and continued fraction type are relatively weak. Their one-side understanding of irrational numbers causes cognitive obstacle in the graphic representation.c. Their performance in the identification of various numbers across sets is less than ideal.d. Their knowledge of the bijection between real number and the real number line stays only in the memory level, which causes them to make wrong judgments on whether a number can be represented on the number line.e. Their intuitive knowledge of "any two segments are commensurable" causes cognitive obstacle in the understanding of irrational numbers.f. In the comparison of rational and irrational numbers, their intuitive knowledge to some extent helps them make the right judgment, but the weakness in formal knowledge leads them to make mistakes.Some suggestions are given at the end of this paper.