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I begin my sessions with some small talk about school, sport or hobbies, depending on the student, as I want them to feel comfortable before we begin. I then ask if there is anything they would particularly like to work on and encourage them to think of topics in advance. This might be something new they have started in school or an older topic they are struggling with.
If they have nothing specific in mind, I keep track of the topics we have and have not covered. I can then return to an area where they previously had difficulties or introduce a new topic that follows naturally from what they are currently learning.
During the session, I like to work through examples progressively, giving the student an increasingly large role in solving each problem rather than simply showing them the solution. I try to create an environment where wrong attempts are welcomed and students are comfortable having a go. I also make a conscious effort to bring enthusiasm to each session, as I think a tutor's attitude can have a large influence on the student's own enthusiasm and confidence. I always give students credit for trying as well as succeeding; mathematics can be difficult, and I respect the effort it takes to work through it.
At the end of each session, I discuss with the student how they felt it went and, where appropriate, speak briefly with a parent or guardian about their progress.