Buying a scorecard at a major league park and tracking each pitch or batter is a rite of passage in American sports, with some fans scoring every game they attend. A similar tradition never developed in football. I'm guessing that was the case for several reasons. Early football was a less discrete game than it is now in that teams quickly lined up after a man was downed and ran the next play, so the game flow was more continuous.
Football players also did not consistently wear numbers front and back until the 1930s, which is when the NFL and NCAA first tracked game statistics consistently. Player numbering allowed statistics to be more easily tied to specific runners, passers, and receivers, but fans could have tracked the progress of games at a team level. The game diagram below shows the ball's movement in the 1896 Cal-Stanford game, a system newspaper reporters used for decades to track the game and was reproduced in their game coverage the next day.
Periodic attempts were made to encourage fans to chart plays, including the 1926 Chicago Daily News Football Visualizer, which I wrote about previously.
Still, I don't recall coming across an example of a fan scoring a game until I found a 1939 Travelers Insurance advertising premium that included formations, schedules, and a blank "Score Chart" to track an upcoming game, though it wasn't blank by the time I found it.
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