I gave my 8-year-old daughter the first tiger-and-the-lady problem from Smullyan's book The Lady or the Tiger?" and Other Logic Puzzles. I turned it into a problem of boxes -- I found two boxes, put a dollar in one box, put the same-sized piece of paper in the other, and taped them shut. On one box was the sign "This box contains a dollar. The other box contains just a piece of paper." The sign on the other box said "One of these boxes contains a dollar. One of these boxes contains just a piece of paper." My daughter was told that only one sign was true and I didn't remember which one. If she opened the right box, she could keep the dollar. If she opened the wrong box, she had to go without candy for a week.
She picked up the boxes. She was very interested. She likes anything new. I said she could bring them to school and get help from students.
She left them home. They sat, ignored, for 5 days. Maybe I shouldn't have tried to compete with the butterfly my daughters had just found, but I was too excited about the problem. Or maybe she just didn't get it. I had never given her a logic problem. She clearly could not do the problem and had no idea of how to think about the problem. In retrospect, I could have found easier problems to start on. But I had already made my boxes.
About 5 days later she came to me and said that she thought the dollar was in the gold box. (The boxes happened to be different colors.) I asked, "So which sign is true?" She said they both could be true. I said that only one was true. She thought about it a little and said, "If only one is true, then the dollar has to be in the white box." I said, "So open the white box." She had to verify that only one sign was true -- she didn't trust me. Then she opened the white box and found her dollar. She told her mother how she knew it was the white box. The little person inside me started a wild dance of celebration.
Solving her first logic problem like this was a huge conceptual leap. I could have tried to lecture her on those skills, though I doubt that would have worked. But I was in no hurry -- she did not have to learn to solve logic problems any time in the next 5 or 10 years. Anyway, once I had given the boxes, I had committed myself to the problems-first approach -- it didn't seem fair to me that the boxes be a segue into a lecture she didn't want and hadn't asked for. Believe me, at the point that the boxes had been ignored for days, if she had asked for an explanation of how to solve the problem, I would have given one. But she didn't.
She apparently needed information from me to solve the problem. But I didn't really provide information in a straightforward way. I just asked a question -- I asked which sign was true. (I almost always answer a question with a question. I had a very good student who once asked me a question and said, "Dr. Frick, couldn't you just once just tell me the answer to my question?")
My information was given in the context of problem solving. My daughter, on her own initiative, was thinking about the boxes. I was fully prepared that my comment might not have any effect. There is no harm in making a comment that is not understood. In fact, I was very surprised that it worked, and worked so quickly.
Third, it was also a thoughtful comment. I had thought about the gap to solving the problem, and my question to her, though constructed on the spur of the moment, started to fill in that gap. Again, the more you know of math, the better you can teach it. But without any knowledge you can still be successful with the problems-first method.