Fuzzy Logic in the APOS/ACE Instructional Treatment for Mathematics

In this paper principles of fuzzy logic are introduced for comparing the performance of two student groups concerning the comprehension of real numbers in general and of irrational numbers in particular. The first group was taught the subject in the traditional way (control group), while the APOS/ACE instructional treatment was applied for the second group (experimental group). The two groups are represented as fuzzy subsets of the set of the grades (from A to F) achieved by the students in a pre-instructional and a post-instructional test and the centroid defuzzification technique is applied on comparing their performances. The results of our classroom experiments show that the application of the APOS/ACE approach can effectively help students to enlist the real numbers in a powerful cognitive schema including all the basic sets of numbers.