Problem:
1. Shuffle a deck and place one card face down on a table.
2. Ask your child to turn the card over. If it’s a diamond, he gets to keep it. If it’s not a diamond, you get to keep it. Continue playing for 10 rounds and see who has the most cards after play has concluded.
3. Ask your child if the game is fair? Why or why not?
4. How can the game be made fairer?
K – 3 children from Trafalgar Elementary School worked on the above stated problem.
We ran into several issues right from the start. We were talking in a language that our students were unable to connect/relate to. I think that we were not using child appropriate language when describing and posing the problem. First, some of the terms, such as dealer, player, suits of a deck, round, fair, etc., appeared alien to many of the students in this context, even though a majority of the students knew how to play a game or two with a deck of playing cards. We repeatedly heard the phrase, “that’s not fair,” despite taking turns when dealing the cards, following the rules of the game, and winning regularly when dealing the cards.
Second, many students seemed to have a hard time trying to count (counting one card twice or not counting at all) the number of cards. This may be partly because the decks were slippery and the cards were too small for the children to handle. On several occasions, students ended up miscounting because of this.
Last, the decks were not sufficiently shuffled from time to time, which led to the player winning a game as opposed to the dealer winning the game. This happened with the first group of students. We didn’t elaborate why this was a possibility, because we thought that the students would not be able to understand the underlying concept of measuring fairness in probability and also due to lack of time. Indeed, when a player had won a game as opposed to the dealer (the dealer had a higher chance of winning a game) winning the game, we did not know how to explain what had just taken place to these students. In these instances, we ended up modifying the game in order to reduce further confusion. Instead of talking about fairness of the game or odds of winning or losing, we ended up steering away from these discussions. We asked students to come up with strategies so that no one would be able to win a game (indirectly bringing in fairness) if the whole deck was played. This modification created its own set of problems. For example, instead of playing a card game of some sort, some students ended up trying to divide the pack into two equal parts and then attempting to count the number of cards.
Winning or losing a card game appeared to be much more emotional to some students. A student who has won/lost the first round often guessed that he/she would win/lose the second round. I don’t know if this observation has any significance, but the body language of the losing player didn’t look too impressive until he/she won the next game. We also noticed that some students didn’t want to lose a game at all. When they did lose games, some students appeared visibly upset. Losing a game, regardless of whether they had won before, appeared to generate a broad range of emotions in some students that seemed to hinder full participation in follow-up activities. Some of them either threw their cards on the table, jump around, briefly walk away from the table, etc. This was noticeable especially when the new winner publicly displayed his/her emotions of winning by fist pumping or taunting the losing player. We tried our best to make sure that every student won or lost the same number of times before moving on to the second part of the problem.
Despite our best efforts to equally share students’ wins and losses, we noticed that a small number of students only wanted to win. These students not only wanted to win every game, but they also needed to win. Losing a game of cards seemed to visibly upset them.
I am sure that fairness is something that many students would have learnt or experienced in their daily experiences. Whether fairly sharing a pizza at home or things with others in their classrooms. In this particular context, it was hard to get some students to experience and understand how fair or unfair this game was. I am not sure if the gym played a role in bringing out the competitive nature of some of these kids. If I were to meet one of the student who said that, “that’s not fair,” I would want to get down to that child’s level of thinking and find out the reasons for this student to come to this conclusion. The game was definitely unfair to the player as opposed to the dealer. A follow up question would be to ask this student to come up with another game so that their game is fair. At his/her level of understanding, I think that this approach could also allow this student to engage with and experience the concept of fairness of games in general.
At the secondary level, this game could be used to analyze and investigate the probabilities of various outcomes. Secondary students would have prior experiences (flipping coins or card games) with the notion of fairness when it comes down to certain games or events. These experiences could be used to calculate probabilities, predict results, and develop algorithms or conjectures about experimental and theoretical probabilities when playing games like this one. Looking at probabilities through a theoretical lens will surprise many students, partly because many outcomes will seem to be counter-intuitive.