Using eye-tracking to assess the application of divisibility rules when dividing a multi-digit divident by a single digit divisor

[摘要] English: The Department of Basic Education in South Africa has identified factorisation as a problem area in Mathematics for Grade 9 learners. Establishing the foundation for factorisation begins at earlier grades. If learners know the divisibility rules, they can help them to determine the factors of numbers. The divisibility rules are presented to learners in Grade 5 for the first time. When a true/false question is used to assess learners' ability to determine whether a dividend is divisible by a certain divisor, the teacher has no insight in the learners' reasoning because he or she is only in possession of the final answer, which could be correct or incorrect. If the answer is correct, the teacher does not know if the learner (i) guessed the answer, (ii) correctly applied the divisibility rule, or (iii) incorrectly applied the divisibility rule. To improve the credibility of the assessment, learners can be requested to provide a reason for their answer. However, if the reason is correct, the teacher still does not know whether the learners correctly applied the divisibility rule �?regardless of whether the answer is correct or not.A pre-post experiment design was used to investigate the effect of revision on the performance of learners and also the difference in gaze behaviour of learners before and after revision of divisibility rules. About 1000 learners from Grade 4 to Grade 7 of two schools were assessed by means of a paper-based assessment on their knowledge of the divisibility rules before and after revision. The gaze behaviour of 155 learners was also recorded before and after revision.It was found that revision had an impact on learner performance per divisor for nearly all grades that participated in the test for both schools. The gaze behaviour was measured as the percentage of fixation time on the digits of the dividend. It was found that revision had an effect on the gaze behaviour for learners who indicated the reason incorrectly before revision and the answer and reason correctly after revision. However, revision did not have an impact on the gaze behaviour of learners who indicated the answer and reason correctly before and after revision.It was found that the correctness of the answer did not have an impact on the gaze behaviour (except for divisor 6) for learners who indicated the reason correctly. However, revision had an impact on the gaze behaviour for learners who indicated the answer incorrectly and reason correctly before revision, as well as for learners who had both the answer and reason correctly after revision for divisor 6. This infers that eye-tracking can be used to determine whether the divisibility rule was applied correctly or incorrectly. Eye-tracking also revealed that learners who did not know the divisibility rules, only inspected the last two digits of the dividend before indicating their answer.The study suggests that when a teacher has access to the learner's answer, reason and gaze behaviour, he or she will be in a position to identify if the learner (i) guessed the answer, (ii) applied the divisibility rule correctly, (iii) applied the divisibility rule correctly but made mental calculation errors, or (iv) applied the divisibility rule incorrectly.An instrument is proposed that can be used by teachers to assess learners on divisibility rules where learners only have to indicate whether a dividend is divisible by a divisor. Eye-tracking will predict whether the learner knows the divisibility rule. For 85% of learners who provided the correct answer, their gaze behaviour corresponded with the reason provided.The study concluded, therefore, that eye-tracking can, to a large extent, correctly identify whether learners, who indicated correctly if a dividend is divisible by a certain single digit divisor, applied the divisibility rules correctly.

[发布机构] University of the Free State