That mechanic can be implemented without starting a new game at all. Increment a counter on correct guess, reset to 0 on incorrect, continue on through the game.
"streak" as i am using it (as it is used in baseball) means "all correct, but for how long", so the proportion correct is always 100%. a "three game winning streak" means you won 100% of the last three games, and 0% of the one game before that.