Simply put, the problem presents a square grid that is 36 by 28 squares. One takes a ruler and starts in the upper left corner, then draws a line at 45 degrees to the outer edge. Here, the line "bounces" at 45 degrees and continues on to the next outer edge. This process is repeated until the line ends up at a corner. The question asked is how many squares did the line cross through to get to the corner. The scale of the problem is large enough to make even the simple solution require some time to get. In addition, several examples were provided that hinted at patterns in the lines that could be used to reduce the magnitude of the problem.
I began by providing each student a print out of the problem statement (minus the question), a sheet of graphing paper, and a ruler. In each class I had two students read through the problem and then conducted my simplified version of Noticing and Wondering. As previously detailed, this involved asking the students to point out what they thought was important in the problem statement, I'd then highlight the relevant passage on the overhead. Quite a few noticed the recurrence of what they called the "fish" pattern and the "upside down heart" pattern. I then presented the question to them, and asked that they try and work through the problem themselves. With about 5 minutes left in my discussion, I called the students' attention back to the front in order to discuss their solutions.
The simplified Noticing and Wondering worked about as well as I could have hoped with several students providing input. Usually singling students out to provide some kind of response is asking for the silent treatment, but in this case most were glad to respond. Even with some modest guidance and hints no one appeared to make the connection that the larger problem could be simplified. As usual, there were minor issues as I struggled to write on the board and projector simultaneously, but this will hopefully be solved with further practice on my part. While I understand the great value in group work, past experience suggested I'd have more results if I asked the students to work individually and Molly agreed in this respect. Mary Beth had come along to observe the class, but got swept up with Molly and I as we walked around and helped the students work on the problem. Surprisingly few were able to follow the rules in the problem statement, and only one or two came up with the final solution. Many had problems understanding the importance of making each line precisely 45 degrees. I made an effort to highlight this fact in the second class with an additional diagram, but I still noticed many students were being a bit impulsive in drawing their lines.
It was interesting to note here that the two classes acted very differently when set off on their own. The first class worked quietly for a few minutes before devolving into commotion and tangential discussion. A few continued their work in spite of the disturbance, but at this point the class appears irretrievable. The second class worked for almost the entire allotted time, and those that didn't work quietly minded their own business. In both cases it took a while to get the students started, but the fact that the problem could readily be solved with some effort definitely appeared to have some appeal.
I wanted to allow the students long enough to reach the solution, but unfortunately my time was limited to 20-25 minutes in both classes. There were no moans of inadequate time when I called the attention back to the front, but I do think that they might've felt cheated. I walked each class through the solution by reducing the problem slowly, and at each step asked the students careful questions so they could lead the way forward. I never did make it to some of the other more interesting solutions, but I felt like those that listened at least had a good starting point.
Ultimately, the problem was a mixed success much like each one before. It is apparent that the patience and work ethic of the students has improved by leaps and bounds over the beginning of the year. At the same time I feel like they do not have a lot of the intuition necessary to solve even the simple problems. In my preparation for the lecture, I hadn't even considered that they would have such trouble drawing 45 degree lines. Mary Beth noted that there didn't seem to be enough time to work the problem out properly. I don't necessarily agree with that characterization. The time was limited for the second class because the majority were actually engaged in solving the problem, but the time allotted was far too long for the first class that lacked the discipline to put more than a passing moment's effort into the work. I had hoped that scheduling my days for Thursday and Friday would help to alleviate the problem of cramming an entire complex problem into twenty minutes, but the scheduling issues have proved to be greater than anticipated. I'm more convinced that the original vision of doing the problems on a day to day basis is the right approach, despite the interruption in continuity but in my position I'm simply not around to make this happen.
This very well may be the last Math Forum problem I do this year so I'd like to make a few notes on the overall experience. The library of problems is immensely useful and well-organized though some have more support materials than others. While most are rooted in real-life applications of math, they still seem a bit off the mark for the students. That is, the students may like football, but that doesn't mean they'll necessarily enjoy a math problem about football. At this point, I think that outright trickery and deception is required to teach mathematics. The problem solving process that went along with the Math Forum problems is very logically set out, but seems better suited to classes with superior behavioral characteristics than mine. Only a small subset of students would engage in Noticing and Wondering, and many times the students seemed to be unclear about the point of the whole exercise. I admit that some of the issues lie with the instructor rather than the students. I could have been more clear in my explanations and instructions to the students. Of particular note, my time management could be greatly improved. While I think there is potential for the program, it really should be implemented on a daily basis and ergo by the primary teacher.
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