Towards Designing Optimal Individualized Placement Tests

To find the current level of a student’s knowledge, we use a sequence of problems of increasing complexity; if a student can solve a problem, the system generates a more complex one; if a student cannot solve a problem, the system generates an easier one. To find a proper testing scheme, we must take into account that every time a student cannot solve a problem, he/she gets discouraged. To take this into account, we define an overall effect on a student by combining “positive” and “negative” problems with different weights, and we design a testing scheme which minimizes this effect. 1 Optimal Testing: Formulation of the Problem Need for a placement test. Computers enable us to provide individualized learning, at a pace tailored to each student. In order to start the learning process, it is important to find out the current level of the student’s knowledge, i.e., to place the student at an appropriate level. Usually, such placement tests use a sequence of N problems of increasing complexity; if a student is able to solve a problem, the system generates a more complex one; if a student cannot solve a problem, the system generates an easier one – until we find the exact level of this student. After this, the actual learning starts. A seemingly natural idea. A natural tendency is to speed up this preliminary stage and to get to actual leaning as soon as possible, i.e., to minimize the number of problems given to a student.