Thursday, September 25, 2014

Teaching and Technology

      I have never been the biggest fan of technology. When I was younger my siblings would play video games or computer games, while I only turned the computer on to write stories or play solitaire. As I grew up this stayed about the same. I got my first phone sophomore year of high school and then never really used it until the end of senior year (my friends would call the house because that was the more reliable way to reach me). It's not that I don't appreciate technology or am bad with technology; I can learn things pretty quickly. I've just always had a slight disinterest towards it.

    Thinking about the use of technology in my future classroom, online activities and web-based programs, I leaned toward the side of no. I feel like there is a lot to said for doing things by hand. But the world is changing. While teachers of the past did not have access to such things, I will. Using sites like Desmos and programs such as Geogebra have shown me technology has a lot to offer a math classroom. Students can experiment with graphing equations, how to reflect/translate/transform them. Data collection can be increasingly simplified. There is an abundance of resources for activities and projects. Technology has the ability to help introduce students to new, difficult to grasp ideas, providing a transition into the topic. Students are able to make references to real situations and see the actual motion a graph is depicting.

   So my mind has changed. However, along with this desire to incorporate technology, which can do amazing things, I still have some hesitance. Yes students are seeing ideas in new ways, and yes they can work at their own pace, and yes they get the chance to explore and discover on their own. Yet, if not careful, the message and ideas you are hoping students to see will become lost. Some of the activities - and I know I have not seen them all - were fun and provide a good introduction to a topic; however, they felt semi-easy. I did not feel I was presented with a challenge or something asking me to think deeply and critically. And this is what I think is truly important. I see technology is helpful, but without in-depth discussions to follow up or asking students to create something of their own (to work through the process of solving a complex problem), it is not going to be used at it's full potential. We need to use technology to challenge our students, not just simplify how things can be done.

Tuesday, September 9, 2014

Eleusis: Problem Solving

The Problem

 Given the following Eleusis card set-up, find a rule to describe it. Then, list three more not yet played cards that can follow the set-up.

The Process

   My first initial thought when viewing the Eleusis card set-up was that color was not the sole factor of the rule. There were two blacks, two reds, black, red, four black. This did not appear to be any sort of pattern. Next I noticed how color affected cards of the same number being right and wrong. I found these four instances:
  • A black 10 could not follow a black 8 but a red 10 could.
  • A black 2 could follow a red 3 but a red 2 could not.
  • A black 6 could follow a black king (or 13) but a red 6 could not.
  • A black 8 could follow a black 7 but a red 8 could not.
   After discovering this, I was able to eliminate adding/subtracting of the cards to be part of the rule. Color would not affect the resulting sum or difference of the two numbers. Getting on the thought of addition and subtracting, I decided to look at the differences between the cards. At the time, I was thinking anything I discover will be helpful - whether I find a pattern or not. As a result I ended up with the following:

   In hindsight, I discovered what I already knew: that the difference between the numbers is irrelevant. Otherwise, if this was untrue any 10 should have been able to follow the 8 because the difference would still be the same. However, I stumbled upon something important. What is the value of the ace? In this rule, was the ace being viewed as 1 or 14, odd or even? To figure this out, I started to propose guesses as to what the rule may be.
  • If you go from even to even, you must change suite.
This instance only occurred once. So I ruled it out and reformulated the hypothesis.
  • If you go from even to even or odd to odd, you must change suite.
There were however cases not covered by this rule, so this was not it either. For example it could not explain why the 2 of hearts could not follow the 3 of diamonds. While neither of these were the rule, thinking and testing my guesses was helpful. I saw what did not work and began to notice my focus on odd/even and the suite/color of the card. So I made the following chart (assuming the ace is 1):

   This simplification of the information given, brought me to a pattern and a rule to describe the Eleusis set-up

The Solution

   The rule for the pictured Eleusis is: An even card must be followed by a red card, and an odd card must be followed by a black card. So based upon my rule I can play the following 3 cards:


    Describing my thought process was the easy part of solving this problem. Although, it took an effort to remind myself to record what all I had been thinking. Having stepped back and re-evaluated the problem several times over a couple days and still no answer, I thought to myself: perhaps it is unsolvable. Perhaps the diagram  is too limited to reveal the rule. Unwilling to give up and a strong desire to overcome the problem - it couldn't be that difficult could it? - I continued to search for the solution from where I left off each time. A lot of thinking, as well as various methods, went into solving this one problem. Even when I was not looking at the problem, I tried to think about it. It wasn't so much as arriving at a correct answer. After all, my rule only holds if the ace is 1 and there was no example in the diagram to confirm an even red card can follow an even red card. I kept trying because I enjoyed the complexity of the problem and simply did not want to give up. I wanted to make sense of the problem at hand and I did.

Saturday, September 6, 2014

Eleusis: A Card Game

Eleusis is a fun and challenging card game that at it's heart encourages critical thinking and problem solving. Always having been a fan of card games and puzzles, I was excited to learn how to play.

The Rules:

Each Player is dealt 10 cards, expect the dealer (so the game is best played with smaller groups).
The dealer is then left in charge of coming up with a rule that the cards played must follow. The remaining cards are sat in a pile. The dealer will flip the first card over, then the players will take turns placing a card from their hand to test their hypothesis of what the dealer's card rule may be. If the players card does not follow the rule, it is placed beneath the card it could not follow. The player must then draw a card. 

The Object of The Game:

To guess the dealer's rule! After each correctly played card, a player can take a guess as to what the dealer's rule is. Once they get it, they have successfully won the game. From here you can pick a new dealer and the game starts over.


The first card played in the example is the 5, followed by the correctly played 7. The 10 was played after the 7, however, because it was incorrect it is placed under the 7. Then the 2, followed by the are played after the 7 correctly. In this example the rule is very simple: Red, Black, Red,.....

Helpful Hints:

  • Don't make the rule too complex. If your rule is too complex it will restrict a lot of cards from being played. The players will likely be unable to solve your rule and there's no fun in that.
  • Don't make the rule too simple. The players will guess the rule quickly and the challenge of the game is lost.
  • Play wrong cards. When you think you have the rule, play a card that goes against it. If the dealer says it works then your initial guess isn't quite right yet.
  • What's the Ace? The ace can be viewed as high or low, 1 or 14, odd or even.
  • Remember the Characteristics. You're searching for a pattern, and in finding this you should remember to consider all the characteristics cards have. Cards are even/odd, black/red, different suites and numbers. Any one of these or a combination can go into the rule.

    While playing Eleusis you get caught up in the game, you forget that you're thinking deeply and critically about each card that is played. Whether a card is right or wrong, you are taking it into consideration. You take multiple steps to try to solve the problem, testing your hypothesis, reformulating it and testing it again. In your head you are adding numbers, subtracting them, analyzing the relationships. Yet, Eleusis is just a card game and you are having fun. After playing the game in MTH 229 (Mathematical Activities for Secondary Teachers) and discussing our thought processes, it does seem like a great game to play in the classroom. Since it is a card game, negative perceptions and connotations of math disappear for the time being. At the same time, students gain and engage in mathematical thinking.

Check out my follow-up blogpost (Eleusis: Problem Solving) where I discuss my thought process in solving the rule for the below Eleusis card game. Try it for yourself and compare results.