As robots are entering into educational fields to enhance children's learning, it becomes relevant to explore different methods of learning in the area of child-robot interaction. In this article, we present an autonomous educational system incorporating a social robot to enhance children's handwriting skills. The system provides a one-to-one learning scenario based on thelearning-by-teachingapproach where a tutor-child assess the handwriting skills of a learner-robot. The robot's writing was generated by an algorithm incorporating human-inspired movements and could reproduce a set of writing errors. We tested the system by conducting two multi-session studies. In the first study, we assigned the robot two contrasting competencies: 'learning' and 'non-learning'. We measured the differences in children's learning gains and changes in their perceptions of the learner-robot. The second study followed a similar interaction scenario and research questions, but this time the robot performed three learning competencies: 'continuous-learning'; 'non-learning' and 'personalised-learning'. The findings of these studies show that the children learnt with the robot that exhibits learning competency and children's learning and perceptions of the robot changed as interactions unfold, confirming the need for longitudinal studies. This research supports that the contrasting learning competencies of social robots can impact children's learning differently in peer-learning scenarios.