Everyone learning to ride at school is a nice idea but practicalities may well get in the way. Not every child has a bike, those that do may not have it in roadworthy order (how long do you keep the good bike kids hanging around fixing cruddy brakes, replacing bald tyres?) others may not want/be allowed to bring their own property to school for compulsory PE lessons.
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I'm also sure chess and lacrosse and basketball and fell running and etc etc etc forums could make viable arguments for their particular interests to be essential additions to the PE calendar too.
I was thinking along the lines of schools taking kids to the local baths for swimming lessons without needing to own a pool themselves.
My cycle training was done at a Butlin's holiday camp when the school took us all on 'holiday'. The bikes were random and we took which ever one fitted best and then the trainer adjusted each for a proper fit. We were then shown what and how to check the bike was safe, the problems with faulty brakes and loose saddles/bars, etc. and then how to ride safely. That included mounting the bike, checking for traffic, pulling away, signalling, cornering, stopping, and so on. Then we could ride the bikes all over the holiday camp for the rest of the day.
There could be ways if the desire was there to do it.
Other sports and activities could still be included in school PE if there is a demand and the facilities available, no reason to not include cycling in that if it were practical. None of the sports are 'essential' really, so long as there are some range of activities that all students could participate in.
My school, for a while, offered only football and cricket (with table tennis when the weather was bad), and swimming only in the first two years as 'learning to swim' classes. Myself and other 3rd+ year students, who had no interest or skill with any ball sports, petitioned the head to give us a 'non ball sport' option and so we had swimming as the option.