During a Math Institute put on by the Public Education & Business Coalition (PEBC), we were asked to monitor our thinking while working on the Coffee Break problem from (warning, this link includes the answer) Discovery Education. Essentially, we were creating a Metacognitive Memoir that would make our thinking, not just our work, visible for others to see. Being familiar with this form of math writing, I quickly began working on solving the problem.
I started by reading the problem - the entire problem. In the past, there were times when I began working on a problem without understanding it. This lead to "wasting" time on unproductive approaches. I did not want something like that to happen on this problem.
The very first step, "Beginning with a full cup of coffee, drink one-sixth of it," got me thinking about similar fraction problems I have worked on. The second step, narrowed the list of similar problems to those involving mixtures. Fortunately, I kept reading because if I had started solving this problem with those previous approaches in mind I would have gotten off to a "bad" start. This problem was quite different from those I had worked on before.
As I was reading, mental images of coffee cups began coming to mind. At first these were simply generic cups of coffee. But pretty soon I realized that I was going to need a model that could represent the quantities presented in the problem.
In order to make the steps of my thinking visible, I decided to make a series of images, like a story board, that showed the ways I was picturing the different steps. First there was the whole cup of coffee. Then there was the cup filled with five-sixths coffee and one-sixth milk. Next I drank a third of the mixture ...
While I initially took the top third, I quickly realized that the milk would need to be mixed with the coffee. I decided to show this by splitting the cup up vertically. But what was I going to do to show drinking a half of the mixture?
As I was debating how to represent this step, and writing down my thinking about it (I could make a three-dimensional model or split each of the eighteenths in half), the facilitator asked us to stop working on the problem so we could share our results. RESULTS?!?! I felt like I had barely gotten started.
Sure enough, some of the other participants had answered the questions:
- Have you had more milk or more coffee?
- How much of each have you had?
What happened next got me wondering, "Why do I make these things so hard?"
Anyone want to guess what happened (what one of the participants said) and why I made this problem so hard? If you'll put your guesses in the comments, I will update this post with the answers next week.
My theory as to why you "made it so hard" - you're a math teacher, not a math student. Therefore, the problem you were trying to solve wasnt' "how much milk and coffee did I drink," but "how could I show this process to students who probably don't have a clue about solving this problem?"
ReplyDeleteI have no idea "what happened" or what the other participants said, tho' I suspect somebody with a "big picture" mind had a totallydifferent approach to the problem.
Oh... and you were also making a strategy that could have been used for situations that weren't as simple. In this case, tho' the language was a little convoluted, because they were always filling up the cup the rest of the way, you were always adding exactly the fraction of a whole cup as stated.
ReplyDeleteThis construct might lead a person to think that that strategy would work even if more complications were added (you added half as much milk as the quantity you'd just drunk...)
The person most likely to get the right answer to this problem the quickest woudl be the person who hates word problems and just grabs the numbers and adds them. Adding a third, a sixth and a half would get a cup of milk, and you started with a cup of coffee. Done deal. None of this "drawing a picture" silliness needed -- which would reinforce the "I hate word problems -- just grab the numbers" strategy...
I agree with Sioux, more or less. Your reasoning (which, by the way, I followed, thinking I'd do something similar... up until you said there was a quick answer, whereby I reread the question and went 'oh, duh') was a generalizable strategy. If the person did NOT drink the whole thing at the end, merely two thirds, it becomes a rather different problem. (I think. Maybe I'm wrong. I'm tired.) But the progressive strategy would still be a way to reach an answer.
ReplyDeleteThere's also the fact that, when marking this stuff, teachers and professors might have to follow along the logic of other people/students. So there's a natural tendency to want to look at the pieces, rather than the whole. Missing the forest for the trees, as it were. Plus there isn't a great motivation for the "whole" here, at least in my case... I hate coffee, and figure if you have to keep adding milk to the darn thing, you have the wrong beverage - just drink a glass of milk in the first place. Playing with fractions and ratios is the only thing holding my attention.