In my K-12 school experience, math was nothing but a chore, a series of rules, formulas and step-by-step instructions that we needed to learn in order to do the same repetitive math problems. And if we did not follow the steps or memorize the rules, we would get the answer wrong. This is a disservice to students, since this kind of teaching promotes learning by memorization, which is easily forgotten when not constantly in use. (I vaguely recall dreading those Summer Math Packets my elementary school would hand out to keep students from forgetting all the material they learned during the school year.) It's also mindless, since it doesn't require the student to think about math or problem solving on a deeper level.
As a prospective math teacher, this is not the experience I want my future students to have. I want my students to perceive math as a process, an idea which my class has discussed very often. I need to find ways to engage my students that lets them use their math knowledge to problem solve without formulas, and one way to do that is through games. My teaching class has been exploring math games to help teach, and there is one game in particular I would like to discuss and it's called Eleusis. One player sets up a mathematical rule and plays a card that abides by that rule. Then the next player puts down a card they think fits the rule. If it does, the card is placed next to the last card, forming a straight line. If it does not fit the rule, the card gets placed underneath the last card. The rule can be very simple like "only prime numbers" or could be more complex like "all red cards need to be 5 or less away from the previous card and all black cards need to be 6 or more from the previous card." The point of the game is to eventually guess the rule by figuring out the mathematical pattern that the rule displays. To show how this game will help students in math and problem solving, I will discuss how I went about figuring out the rule, which was a homework worksheet Professor Golden provided.
Here is the setup:
The first thing I do is to see if there are any simple patterns based on color, like if all red cards are even and black cards are odd or if all red cards have to lower than the previous card and black cards have to be higher. At first glance this does not seem to be the case. The rule also doesn't stipulate that the color or suite has to alternate in any kind of pattern. Then I try to rule out more simple rules by asking myself "are these all prime numbers?" That doesn't seem to be the case either. So, the next thing I do is to look for more complex rules, like greater than or less than patterns. For example red cards can only be played if it's greater than or equal to 3 but less than 6. That also did not appear to be the case. Then I tried to focus on the suite. For example the rule could be diamonds are prime, spades are multiples of 3, clubs are multiples of 2, and hearts are multiples of 5. This would be a very hard rule since there is crossover between suites. For example, 6 is a multiple of 2 and 3. But again, the rule did not seem to involve the suites. Then, I thought I jumped to exploring complicated rules to quickly before ruling out the simpler ones, so I decided to see if there was a simple pattern between the value differences of each card. Below is my written work for this approach:
I noticed a black card, no matter what suite, could only be played if it was one value away from the previous card. So, I have part of the rule: Black cards have to be +1 or -1 from the previous card. Since, black cards had to be one value away from the previous card, I thought the rule for the red cards was that the value had to be greater than +1 or -1. This seemed to work except there is a 7 of diamonds underneath the 3 of diamonds. Both are red, 7 is +4 away from 3, and yet that card did not fit the rule. Then I thought, "Oh! 2 cards of the same suite can't be next to each other," but this was also not the case since there is a 10 of diamonds and a 3 of diamonds next to each other, and that's a difference of -7. Then, a thought hit me: maybe the rule is if a red card is played and it's the same suite then the card has to be subtraction only, not addition and the difference has to be greater than 1. That rule seems to fit, however if I was playing the next turn I would want to place a heart card next and have it's value be +2 or greater to see if my rule still holds. So, my best guess as to the rule for this pattern is if a black card is played then it has to be +1 or -1. If a red card of a different suite is played then it has to be greater than or equal to +2 or less than or equal to -2. If the red card is the same suite, then it has to be less than or equal to -2.
By explaining my thought process when playing this game, I hope I've illustrated how Eleusis builds problem solving skills and requires the use of math knowledge that isn't a formula or memorization of steps to figure out a problem. To problem solve, students will learn how to use both the correct and incorrect cards to figure out the rule. Looking at one and not the other gives you incomplete data to work with, which will lead to an incomplete rule. The correct cards gives what it could be, but the incorrect cards helps rule out what it can't be, which is more useful in narrowing down the rule. And even when I got to the end of the card play, I still wished I had more information, more cards to compare to check and verify my rule. They will also learn to continue to make guesses, and then test them like I did, a problem-solving process called trial and error, to eliminate all the wrong rules to finally reach the right one. Students will also have to think more deeply about using math to find patterns for the rule they are trying to solve. Formulas to find rules are nonexistent since each rule is different and no game will be exactly the same, even when the same rule is used again, since the cards played will be different each game. So, students will have to start from scratch to figure out the same rule again. All in all, games like Eleusis are a great tool to help students view math as less like a chore and more like a process, since it will engage them to think about math and problem solving more deeply.
***For those who are curious about whether or not my rule was right, it wasn't! The rule Professor Golden used was if the previous card was odd, then a black card must follow, and if the previous card was even, then a red card must follow. I was completely off! But this just illustrates the importance of using the incorrect cards to eliminate possibilities. If I could have played the game further, then I would have played certain cards that would have verified or eliminated my rule altogether. Trial and error and the process of elimination are very important problem solving concepts that this game illustrates!***