Thanks for your letter. It reminded me of one of my favorite Macgyver quotes.
While you might not want to use the flashcards to reinforce false beliefs about doing math, they might serve other purposes.
We could ask our students to sort the flashcards; "Which facts do you know and which facts are you still learning?" Then we could see if there are any known facts that they could use to help them learn the unknown facts.
Students could use an interesting/appropriate subset of flashcards to create a real graph of the answers. They could analyze the graph looking for patterns, making conjectures, and testing the conjectures. Like, "I think that if we keep sorting out our flashcards, 27 will show up the most because that's the largest product in our set of cards."
We could use the flashcards to create Which One Doesn't Belong [WODB] scenarios. Or have students create their own WODB to challenge their peers. (See the WODB blog and the book by Christopher Danielson for more information about this instructional approach.)
Finally, if we want to play a game that's not Around the World, maybe we play Go Fish. (Make up your own rules depending on what you want student to experience. Or better yet, maybe have them make the rules if you want to encourage them to play with math.)
What do you think, regular (irregular, and new) MacGyver Math readers? How might Enlightened Elementary Educator in Elmira step back and take a look at what she's got (flashcards) in a totally different way? As always, leave your suggestions in the comments.