So I’ve been working on the dishes problem, on my own. I’ve found this particular question far more engaging than others, and am questioning why. It’s a question I have previous experience with, and have even given students the first variation during a probability unit. In the last two days however, I have delved into discovering the generalization for the problem far deeper than I typically would before getting bored and have really enjoyed the experience. The feeling of success when I discovered a generalization that holds is one I definitely want my students to experience. I am guessing a few things are to credit for my willingness to see this problem farther than others.
1) As previously mentioned, it is a question I was already familiar with so I came to the problem already knowing what would get me closer to “fair” and what would get me farther away. I’m not sure this is really a prerequisite since this isn’t something that takes a long time to figure out however.
2) Starting this in class with a small group, we discovered a really nice way to represent the possibilities visually that I’d never seen before but that really works for me. This allowed me to continue working on the problem at home in a more efficient way than if I was using a different method. While we did collaboratively discover this way to represent our answers, I definitely wouldn’t have thought of it alone no matter how long I spent on the question myself.
3) Doing trial and error using the method mentioned above was simple enough that I was willing to do a lot of it because I was certain a pattern would emerge eventually. My guesses kept being wrong, but testing more things out wasn’t onerous, just a little time consuming.
Finally seeing a pattern, and being able to confirm that it will hold 100% of the time was awesome. And while I love working with a group (and in fact would likely not have gotten as far without our initial group work) knowing that I did it myself was also very rewarding. Sometimes when you come to a conclusion as a group you’re never really sure who is actually responsible for the final answer (and most times it really doesn’t matter) but it was a great reminder that I can think mathematically on my own and be successful, even if I do take more time to do so.
I’m pretty sure that this is what the new curriculum wants for our students – that they feel empowered and capable of doing math, and that because of those two things they can become deeply engaged in it.