## 1 Introduction
Online formative assessment systems, web-based adaptive learning systems such as intelligent tutoring systems, and e-learning management systems are increasingly of interest in the field of education since the COVID-19 outbreak. The success of an adaptive learning system is fundamentally grounded on accurate information about what a student has learned and knows about specific concepts [15; 16]. As such, a valid and meaningful adaptive learning system necessitates accurate assessment of a student's ability and diagnosing what concepts a student knows and has learned. To achieve this, the adaptive assessment system must be able to interact with the learning system. Computerized adaptive testing (CAT) provides a customized item set by dynamically selecting appropriate items aligned with the learner's ability. CAT has been used for one-time testing such as in selection assessments or certificate exams [17]. With CAT, each item of a test is exposed to a learner depending on their response to the previous item, so the items are adaptively administered to learners in terms of their abilities. Therefore, the items administered to different learners are unique and depend on the learner's ability [18; 19]. Figure 1 shows the procedure of CAT. In order to implement CAT, all items should first be calibrated by item response theory (IRT) [20]. Various IRT models can be used for estimating item characteristics such as item difficulty and item discrimination in order to build an item bank. Once the item characteristics including item features, difficulty, and discrimination, have been estimated using various statistical estimation methods, the items are stored in the item bank along with their information. When administering a test using CAT, the process starts with selecting the first item from the item bank. There are several statistical methods proposed for selecting the first item from an item bank such as a random or Bayesian method. In the next step, an examinee's ability and the standard error of measurement of the examinee's ability are estimated based on their response to the first item using IRT. After this, the next item is adaptively selected from the item bank based on the previously estimated examinee's ability and the computed standard error of measurement. In more detail, CAT selects the item that maximizes the discriminatory and informative values for the given examinee's ability. For example, if an examinee answers correctly, the next item will be slightly more difficult than the previous one, however, if an examinee answers incorrectly, the next item will be slightly less difficult than the previous one. This procedure is repeated until the stopping criterion has been met. The stopping criterion of this iterative process can be (1) a certain level of standard error of measurement for a learner's ability, which is about measurement accuracy or precision or (2) a pre-set maximum number of items [21]. Consequently, by integrating CAT technology into a formative assessment system, the formative assessment can be adaptively implemented in terms of a learner's characteristics. By combining the statistical and mathematical properties of CAT with a formative assessment system, a new assessment system is created that can adaptively measure not only what a student knows, but also what a student's learning progression is over multiple measurement time points. From a diagnostic perspective, it offers information about which parts of a curriculum are difficult for a given student to learn as well as how well the student is doing throughout a course. Furthermore, since all students do not need to answer the same items- rather, an adaptive number of items are administered in terms of their abilities based on the CAT algorithm-, testing can be more efficient, accurate, and informative [22; 23; 24]. Historically, most CAT applications have been used for one-time testing instead of for multiple measurements over time such as in formative assessments.