I'm a pusher
. If I wasn't ds wouldn't do anything and he wouldn't have the enjoyment out of life that he has now.
Take karate - he's been doing it since he was seven (he's now 13) and every single time he misses a training session for holiday or whatever he is reluctant to go back in case he's missed anything. I've had to virtually shoe-horn him back into the hall each time, with promises of rewards and what have you - and each time he's come out beaming because he's enjoyed it so much.
I use that too, remind him how much he enjoys it once he gets there and how he'd miss out on that fun if he just sat and watched tv instead.
Next month he's taking his 1st Dan grading. He wouldn't be if I hadn't pushed and cajoled. He's bursting with pride at being a senior club member
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I think you have to weigh up the pros and cons and decide whether your child would be missing out if they opted to quit an activity, or if they genuinely don't want to do it for their own valid reasons. Only we as parents can know that, but even then it's such a grey area. I have pulled ds out of some things that I knew instinctively were wrong for him, but yes, I definitely push him to those I think he's gaining from, whether he realises it at the time or not.
BTW, totally agree with oodles suggestion of mild bribery to encourage him to keep trying. Have done the same on many occasions, always successfully!