Development of a Web-based Knowledge-State Inference System for Guiding Middle School Mathematics Learning

Abstract

This paper describes a Web-based Knowledge-State Inference System which manages middle school students' problem solving abilities revealed from test items. This program consists of two modules. The first part is a module that draws knowledge inferences for a given student's problem states based on pre-built test item attributes. The second part decides which abilities of the student need complementary practices according to the diagnosis done by the inference system. This second module also provides students with adaptive ways of taking relevant problem items mapped from item-attribute matrix in the domain of middle school mathematics. All of these mechanisms are operated on the WWW environment via Common Gateway Interface (CGI). Our system was designed as an aid for performance assessment which is one of the major evaluation methods in Korea. We begin with carefully examining test items in terms of content domains and student abilities. Content domains are divided into number and expressions, equations, functions, etc., whereas student abilities corresponds to the ability to express words in equations, the ability to apply formulae to solve a problem, the ability to calculate without error, etc. By constructing a binary matrix with content domains and student abilities, each test item can be mapped into a content-ability matrix. With this test item matrix, we proceed to building knowledge-state inference system based on Bayesian Inference Network algorithm. In order to apply Bayesian Inference Network algorithm to our test item matrix, we first figure out the relationships such as dependency among contents and abilities for each test item. These relationships are clearly represented by directed acyclic graph. Thus, all the items are represented in the same way. Certain contents and abilities can be pre-requisites of other concept, say node N1, which means students first need to understand the pre-requisites of the concept (N1). Priors of network nodes are empirically assigned according to teachers' teaching experiences. This big graph is then an input for computer programs that calculate Bayesian inference. We tested ERGO and Netica for WWW environment. The result of executing such a Bayesian program updates probability distribution among nodes. Such probability predictions are discretely scaled so that school teachers and students can easily look up their knowledge states related to problem solving. The configuration of the final Bayesian network also indicates which parts of the student's learning call for extra guided learning. Supplementary learning process is in turn managed from WWW-based instructional module that communicates with Knowledge-State Inference System.