In a recent guest post "Why Khan Academy is not the answer" Daniel Kitts writes: "Popular efforts to improve education are focusing on the wrong problem. Millions of dollars and hours of innovation are being spent on improving how we deliver content in an era when content matters less and how we interact with it matters more. [...] Future education is better served by investing in and developing tools that support discussion and interaction, not improving content delivery."How can I create the conditions within which people will motivate themselves? - Edward Deci & Richard Flaster
A good example of content that will matter "less" in the future is Algebra. That's not to say that Algebra will not be important, it will just be approached differently or so I hope. For those of you that think this is a bad idea, don't worry there will be plenty of students that will continue on the geek track devouring Algebra in its current form. What I'm getting at is that the traditional, step by step approach - as outlined in the table of contents of almost any algebra textbook - will become an option, rather than requirement. Math courses that will be a subsitute for traditional algebra will focus more on real world problem solving, answering questions that are more meaningful to students than learning to solve quadratic equations in isolation.
The heart of effective problem based learning is this: the teacher sets the stage with a problem, puzzle of game containing an interesting context where math isn't mentioned, but when all is said and done the students learn some powerful mathematics.
I learned this lesson in the early 1980s when microcomputers were just beginning to enter the schools. I had a 10th grade high school student - I'll call him Don - who challenged me to respond to his complaint that this Algebra 2 stuff he was doing in my class was a complete waste of time. I sympathized with him. I knew the way I was teaching Algebra made no sense to him. What he did for me was to bring out into the open the conundrum that I faced every day as a teacher: how to motivate a subject that most students see as irrelevant. For them Algebra was something to be avoided unless absolutely necessary (e.g. passing my class) and where a C grade was fine with them. Even when I offered them a blueprint for getting an A students like Don never tried even though it was there for the taking. I offered it up guaranteed if they worked for it. For me an A was short for Achievement (reaching a goal) not an indication of how clever or smart they were. Though Don was taking Algebra it was clear to me he had no understanding or appreciation for anything mathematical that was different from the everyday math that he could handle. (Mostly in relationship to money which he knew a lot about.) If I wanted to have any success with Don I had to dig deeper into his story to find a way for him to feel successful and appreciate Algebra. I didn't need to look at his school history. All I needed to know was that he loved to play Craps the 2 dice betting game. The basic game is played as follows.
The player throws two dice. If the sum is 7 or 11, then the player wins. If the sum is 2, 3 or 12, then he loses. If the sum is anything else, then he continues throwing until he either throws that number again (in which case he wins) or he throws a 7 (in which case he loses). (Source: http://mathforum.org/library/drmath/view/70263.html)Don knew these rules cold. But would he have ever thought to ask: is Craps a fair game? Or what are the chances of winning? Are they 50-50? Did he understand that playing the game over and over again could help him determine whether it was fair or not?
All I knew was that he thought Craps was a fun game. That's all.
So I asked him, "Do you think it's a fair game?" I didn't have to explain what I meant. He understood. "Sure it's fair," he said. "Sometimes I win and sometimes I lose." "But what if I told you that the dice were stacked against you," I replied. "In other words, in the long run you will lose more times than you win."
"How do you know?" he asked.
"Do you think gambling houses play games that are completely fair meaning that both player and the house have an equal chance of winning?" I queried. No response. He was confused. I went on.
"Does a gambling house make money?" I asked. Don replied, "Yes."
"Then they must have an advantage, right?" I continued.
At this point I suggested that we play the game in class to find out if that really is true.
But before we started rolling dice, I wanted my class to understand how probability works so I played the coin flip challenge with them. I split the class into two groups creating two teams. I would play as the third team. Now I tell them that I'm going to flip 2 coins. Team A will score a point if both coins are heads, Team B will score a point if they both are tails. And I will get a point if it comes up a head and a tail. We played a few rounds. It wasn't long before the class realized that I had an unfair advantage since after 10 tosses I was well ahead with 5 points. (A had 3 and B has 2 points.) Why is that? Well, there are actually 4 possibilities not 3 (as was assumed by my students.)
HT and TH are separate events so my probability of winning is 2 out of 4, not 1 out of 3 as I led the class to suspect. I believed that playing this coin game would help my students to understand the dice probabilities better later on.
Now it was time to play Craps.
"I'll be the house and you and the rest of the class will be the shooter," I said, speaking to Don.
A big smile appeared on Don's face. Finally we were doing something in Algebra class that was fun and as far as he knew had nothing to do with math. Or at least that's what he thought.
We had time to play a few games. Fortunately, I had a supply of dice in my desk drawer. We finished playing 5 games when the bell rang. Don and the class won 4 out of the 5 games. Feeling smug Don left the room with a smile on his face. The following day I asked the class if they thought this game was fair? The class had different responses: yes, no, maybe and I don’t know. "How can we find out?" I asked. Don said for us to keep playing. "I bet ya if we play enough games we will win more often than lose." "Ahh, what an excellent idea," I thought. "What does he know about the law of large numbers or Monte Carlo methods?" I wanted to find out.
"But how can we keep playing a lot of games quickly?" I asked rhetorically. "What if I could build a machine that would play the game over and over." "You mean like a computer program?" asked Don. "Exactly," I said. With the help of several of my students we eventually had these instructions written on the board.
|Table 1. To see actual code written in Scratch click here.|
Don wasn’t impressed. He still thought it should be 50-50. How to convince him that knowing that its closer to 49% than 50% makes a big difference in the real world of gambling? Our next conversation would be about that.
My conversation with Don went something like this:
Ihor: How many possible outcomes can you get when you roll 2 dice?
Don: 2 through 12. Let's see.
Then with his fingers he counted. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12.
Don: There’s 11.
Ihor: That's right. Those are all the possible outcomes. But how many different rolls can you have? Snake eyes [two ones] is one roll, Boxcars [two sixes] is another. How many are there all together?
To help them I made a tansparency template with the 6 dice roll possibilities along the top and the left side of the grid and put it on the overhead projector.
|Figure 3: A grid overlay for possible outcomes when rolling 2 dice.|
We filled in the numbers together. Each number inside the grid is the sum of the column and row headings (photos of the actual dice faces.). For example there are 3 ways to make a 4 (3+1, 2+2, and 1+3.)
|Figure 4: All possible outcomes when rolling 2 dice.|
The class got it.
Ihor continues, "2, 3 or 12 on the first roll are losers and since there are 4 ways you can lose the chances of you losing is 4/36 or 1/9. So what that means is that a one-third of the time you will win or lose on the first roll." (2/9+1/9=3/9=1/3)
"But what if you roll a 4?" I asked.
Don immediately responded, "You don’t win or lose. Four becomes your “point” and you roll again. If you then roll a 4 you win. If you roll a 7 you lose. If you don’t get either, you roll again until you get either a 4 or a 7.
"What do you think your chances are of making your point?" I asked.
Don said, "Since there are 3 ways to make a 4 and 6 ways to make a 7, your chances of winning are 3 out of 9 or 1/3."
I was impressed. But I wasn't sure any one else in the room understood. So I said, "Let me see if I can sum so far. The chance of rolling a 4 on the first roll is 3/36 or 1/12. The chance of you rolling 4 again before you roll a 7 is 3/9." Then I took it one step further and said, "Since the two outcomes are independent of each other the probability of you winning when you roll a 4 is 3/36 x 3/9 = 9/324=1/36 which is 2.78%. The same argument applies to the other 5 numbers (5, 6, 8, 9, and 10) that escape the immediate win or lose test.
I showed table 2 to the class.
|Table 2: Probability of winning at Craps (1)|
I replied, "You rolled the dice twice. Were the probabilities of each roll effected in any way?" "No," said Don unsureadly. "If dice rolls are independent you can multiply their probabilities to determine the probability of the outcome." Now Don (and probably everyone else in the room) was really confused. He did with me what he always did in math class he capitulated to my authority by saying "if you say so." The chart claims that the shooter has a little better than a 49% chance of winning, but why?" Table 2 had to be right because Mr. Charischak said so, but the reasoning was just hocus pocus to Don as I'm sure it was to the rest of the class.
"Can we verify this claim about the probability in another way?" I asked trying to relieve them of their math anxiety. "No, its not going to be on the test," I said answering what I'm sure what was on their mind.
"What do you think will happen if we run the experiment 1,000 times?" I asked. "We're going to beat the computer," was the reply meaning that the shooter would win. I estimated it would take about 2 hours since the output was going to the printer so we waited until the following day to check the results. Don was pleased to see that “the shooter” won 511 times. "Does that make the math [in the chart] wrong?" I asked. The responses from the class were split between yes and no. "What do you think will happen if we run the simulation 10,000 times?" "I don't know," said Don. "But let's try it and see." I reprogrammed the simulation to do the additional rolls and I also had the computer print the results to the screen instead of the printer. I don't remember the exact results but the shooter was the loser. We then talked about the law of large numbers and how the experimental results came closer to the actual probability because of the large number of rolls.
For fun I ran the experiment last night using a program I wrote in Scratch and it approximated the actual probability (49.29%) well. 4,961 wins in 10,000 games. This time I smiled because it confirmed the mathematical analysis.
After this experience I vowed that Craps would became a permanent part of my Algebra “curriculum” without going into the details of determining the theoretical probability. Just highlighting how the computer program uses variables to produce its results was a dynamic example of using variables in Algebra. Unfortunely, I never got to write it since my days at BFS were numbered and I was going to move on to other adventures. Looking back I realized that the the seed of what I now call the Wannado curriculum were planted firmly, but making it happen was another story.
Don and I had a productive learning experience. Don learned how to use variables in programing the game which we wrote together. He understood the "law of large numbers" after we ran the simulation a 1000 times. The light bulb went on when he realized how the casino had the financial edge so playing Craps in the long run was a losing proposition. The key to the success was that I was a student, too. I was in the beginning stages of learning to program myself. When he realized he was my partner in programming the game I saw the sparkle of engaged learning in his eyes.
Years later I learned from Elliot Soloway (2) that he did research in how programming supported students ability to problem solve. For example, he gave this problem to college students to model algebraically: "There are six times as many students as professors at this university. Use S for the number of students and P for the number of professors." The most common answer was 6S = 1P with the students thinking that this equations means there is a large number of students associated with each professor. Then he asked the group to write a simple program that would take as input the number of professors (P) and output the number of students (S.)
They realized that the number of students S was equal to 6P not the other way around as previously thought. So programming a computer helped the students realize the correct use of the variables involved.
1. The original source for this chart is long since forgotten. You can s a similar chart here: Smith, S. (2009) Craps ... A Casino Game of Pure Chance.
2. Soloway, E., Lochhead, J. and JohClement. Does Computer Programming Enhance Problem Solving Ability? Some Positive Evidence on Algebra Word Problems. In Computer Literacy, edited by Robert J. Seidel, Ronald E. Anderson, and Beverly Hunter, 171-185. New York: Academic Press, 1982.