The Tottenham Hotspur forward fell to the ground after Alves made contact but immediately got back to his feet and started chasing the ball.
The challenge earned Alves a straight red card and Hodgson said Kane's behaviour exemplified English football culture -- something the coach is not willing to meddle with.
"Harry's first instinct, when he didn't get kicked severely, was to carry on and do something with the ball," Hodgson told British media.
"Some people might say that's very laudable, others might say 'You've got to go down, you've got to be cynical'. But I find that cynicism quite a hard thing to coach.
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"I think it has to be taught, if it's going to be taught, at a very early age to be part of your culture and I've said many times that I don't think it is part of our culture.
"There will be occasions -- and I take the point -- when there will be a (possible) penalty but the player stays on his feet and maybe then it will be a very relevant question.
"But again, all I can say is that it's hard for me, being English, to start trying to teach people a manner of playing which I've never subscribed to and they don't subscribe to."
England have been drawn in Group B and face Russia in their Euro 2016 opener on Saturday. Slovakia and Wales are the other teams in the group.
(Reporting by Simon Jennings in Bengaluru; Editing by John O'Brien)
