# Computer Adaptive Testing Using Upper-Confidence Bound Algorithm for Formative Assessment
## 1 Introduction  
Formative assessment have been proposed to make education more accessible and more effective [1; 2; 3; 4]. The distinction between summative and formative roles of assessment was first proposed by [PERSON] [5] and then applied to students by [PERSON] [6; 7]. Formative assessment is specifically intended to generate feedback on performance to improve and accelerate competency acquisition as opposed to summarizing the achievement status of a student [8; 9]. Any learning activity has potential value as formative assessment from oral discourse to conventional quizzes [10]. Three core principles form the basis for formative assessment [11]. Firstly, formative assessment should be viewed as an integral part of instruction, and it should be used in real time for guiding learning process. The material provided to students should depend on their current state of knowledgeand understanding. Secondly, formative assessment fosters student involvement. Students are not punished for their mistakes since assessment does not affect the final grade. Instead they can use assessments for self-guidance and to monitor their progress towards learning objectives. Thirdly, formative assessment requires constructive feedback. Feedback is the key element in formative assessment [12], it is claimed to be the single most powerful factor in enhancing achievement [13]. Studies indicate that feedback may improve learning from about 0.4 Standard Deviation (SD) to 0.8 SD [14], however a critique of these results exists [2]. One of the goals of educational process is identifying the gaps between what is known and what is aimed to be known [15]. Feedback which is separated from competency demonstration by days or weeks, as it happens with paper based tests that have to be manually graded, have diminished value [10]. Feedback in a form of a mark as provided by traditional tests has a limited use to a student as it is not congruent with good feedback practices [16]. Simply put, a mark does not specify the deficiencies in the student's knowledge which they should address in the future [17]. Another strength of formative assessment is that it can aid in using students' strengths and weaknesses to frame learning goals and monitor progress towards them [18]. In addition to linear traditional tests with fixed test items, adaptive tests based on Item Response Theory (IRT) have been proposed [19] and implemented [20]. In IRT computer adaptive tests each test item is dynamically chosen based on the student's answers. This can improve the precision of testing by adapting to the knowledge demonstrated by each student and shorten test times [21]. These advantages come at cost of introducing several new issues. Most problematic of which are greater test creation complexity because of the necessity of calibration of question pool and the premise that different questions measure one common trait [22]. This premise is not always useful when assessing separate competencies even in the context of Multidimensional Item Response Theory (MIRT) [23], which can even further exacerbate the challenge of item calibration. In practice, the main drawbacks remain the complexity and time required to create a test, which may dissuade teachers from using it as a learning tool. Another alternative to traditional methods is Elo rating based approach [24]. Elo rating system was originally developed for chess skill estimation and ranking. But since has been adapted for use in education [25; 26]. Within the system each participant is assigned a numerical rating and in case of educational applications so are the test items. The expected probability that a player wins, or a student successfully completes the task is then given by the logistic function and depends on the ratings involved. In this formulation Elo rating system estimates the probability of correct answer in the same way as one parameter IRT model (Rasch model) [27]. What differs is the parameter estimation procedure. Analogically with MIRT multivariate Elo extensions have been tested [28].