Tuesday, March 5, 2013

Quadrillions of Pennies?

     The other day, a colleague of mine and myself were discussing exponential growth and how to open this to his students. I've never taught it before, so I felt that I wasn't the best person to ask but thought I could learn something as well. He suggested the classic problem of 'would you rather have $10,000 now or one penny today, and twice as many pennies each day for a month.' I thought this would be great, and I mentioned the similar problem of if you have a bean on the first square of a checkerboard and you double the number of beans for each square, how many will you have on the sixty fourth square. He decided to go with that instead, but use pennies instead of beans. Long story short, his students were amazed at the fact that there would be 92,233,720,368,547,800 pennies on the last square, not to mention the total amount of 184,467,440,737,095,000 pennies on the board.
     Needless to say, both of us were also shocked. In my geometry classes, I've been trying to take our calculated values and putting them into a context that students can relate to. Saying that a box has a volume of 150 cubic feet means nothing to a student until they can see what one cubic foot looks like. So having a value so large was just that to his students: a large, inconceivable number. We tried to put it into a context that was relatable but we kept coming up short. Nothing that we did, relating it to the dollar amount, distance, etc. made sense to us or our students. We came up with one good comparison, but it still wasn't the best.
     I decided to use this as an opportunity to experiment. As we were discussing the number I had students in my room taking a test. We were speaking just about the numbers but did not mention the context. One student looked at me afterward and laughed saying, "You guys are such nerds, talking about big numbers." I went on and on about how cool it was for the answer to be that big, and then I realized he had no idea what I was talking about. I described the original problem to him, and he became interested. I then put it into context, saying that's like if you stacked pennies on top of each other, that stack would reach Pluto from Earth 1,911 times. His response: "Wow," and then he pondered. I could see the look on his face, focused on what I just described to him.
     Subject #2 conversation went like this - Student: "Mr. B what's this large number on your board?" Me: "Go ask Mr. Miller, he'll tell you." Student comes back: "Yeah, he said something about pennies and checkerboard. I wasn't really paying attention." Me: *describes problem* "That's how many pennies would be on the last square." Student: "Ok." Me: "That's like if you stacked all of the pennies on top of each other, that stack would reach Pluto." Student: "Woah. That's awesome." Me: "955 times!!" Student: *mind blown*
     I type this because context matters. I've tried this discussion with many other students and the response was always the same: they start out not caring but then are super interested when they are able to relate to what is going on. Now, I understand that there are some math problems and topics that can be engaging in the pure form that they are. I'm not saying that everything has to be taught in a real-world context. However, students need to be engaged and they need to have a reason to be engaged. I do not like when entire courses are taught without any context whatsoever. How does this help students? In my opinion, this helps to create students' hatred of math. They don't have a reason to care about it so they don't try, which results in them not succeeding, which results in anxiety, and the cycle continues (well, perhaps that's an exaggeration, but it certainly does not help). I want everything I teach to be engaging and relatable, whether its connected to the real-world or previous topics we've discussed. Nothing should be so abstract that they can't comprehend what's really happening. I believe math can be interesting to all students, we just need to figure out how to get them hooked from the beginning and everything we can to keep them there.


  1. "... we just need to figure out how to get them hooked from the beginning..." Yep! We did this problem in Excel, and same reaction, mind blown. Thanks, Patrick.