Dichotomous Response Testing is widely used in the items of traditional cognitive tests, but examinees often receive the same grades in such items and their abilities cannot be distinguished. Some other tests using Polytomous Response Testing are constrained by the analytical methods and only homogeneous items are used, e.g. multiple choice questions. Hsiang-Chuan Liu (2007a) considered all possible heterogeneous weighted averages systems as well as the standardization and normalization of scoring, integrated the systems with Item Ordering Theory, and proposed the improved theory - Ordering Theory for Standardized and Normalized Polytomous Response Testing Parametric Item Response Theory has the constraint of local independency and can only be used to analyze items without ordering relationship, while Nonparametric Item Response Theory is not constrained and can be combined with the Item Ordering Theory. Ramsay (1991) first applied Kernel Smoothing Nonparametric Regression Model to Nonparametric IRT Models for dichotomous response testing, and introduced “Kernel Smoothing Nonparametric IRT Models”. Hsiang-Chuan Liu (2007b) further used the weighted averages of the product-moment correlation in Polytomous Response Testing to substitute that of the point-biserial correlation in ichotomous Response Testing, and proposed a expanded, improved model - Kernel Smoothing Nonparametric IRT Models for Polytomous Response Testing. Combining with “Ordering Theory for Standardized and Normalized Polytomous Response Testing”, he published “Kernel Smoothing Nonparametric IRT Models for Polytomous Response Testing”. This study applies “Kernel Smoothing Nonparametric IRT Models for Polytomous Response Testing” to develop programs. An empirical study was also conducted by using the unit “The Addition and Subtraction of Fractions” in fourth/fifth-grade math to design written tests. This study concludes that: I. Out of the 122 effective samples, only 16 examinees receive the same grades through Polytomous Response Testing based on standardization-normalization scoring. None of them receive the same grades if the weighted averages of the product-moment correlation are used. This model can distinguish the examinees’ grades and increase the degree of differentiation of the tests. II. The possibility for a question correctly answered is higher with the increase of the examinees’ abilities. For examples, the 11 questions relative to the subtraction of fractions in this study concludes that .0468 - .6750 stands for low ability, .6080 - .9495 for average ability, and .8845 - 1 for high ability. These can be useful reference when teachers are examining the examinees’ performance. III. The item ordering structures differ when the abilities are different, showing that examinees with different levels of abilities differ in their conceptual structures for the subtraction of fractions and in their path of learning in this unit. Teachers can thus plan the order of teaching and provide individual remedial teaching.
