What task did you feel most successful with? Why?
"Problem A...The way this problem is designed, there are several different ways you can approach it."
"Problem A...we definitely tore this problem apart. We looked at all the different possibilities, all the restrictions, we looked at a variety of theories and hypothesis, and we proved or disproved them. I felt successful because we created rules that we could use for any given triangle and it really helped me to understand the problem more. B and C were good, but they were more, like, content-based I would have to say. But with problem A, there was no real content idea that you had to learn."
"Problem B...my main problem is I never know where to start and with this problem we were kinda given a place to start."
"Problem A...Unlike B and C, I was able to get started on this problem right away. With A, you can explore in whatever way you want to, so you are almost, like, being successful the whole time because there isn't really a right or wrong, you are just testing.
Which task did you enjoy the most? Why?
"Problem C...I enjoyed this because it was really visual and you could clearly see the diagonal lengths."
"Problem C...it was outside of my comfort zone. Even though we used Pythagorean Theorem it was still challenging and a new learning experience."
"Problem A...It bothered me that I didn't actually know how to start. It was fun to play around and see what you could do but it was like 'frustrating fun.'"
"Problem C...before this problem we were working on the Geoboard and playing with squares and Mr. Meyer came over and showed me a diagonal square. I had never even thought of that and it just opened my eyes and I started to enjoy it and I started just looking for all these crazy squares." Here he is actually referring to a problem we did before Task C.
"Problem B...I enjoy structure a lot. With Task A we were just drawing a bunch of triangles and my mind was, like, 'what?!?' were are the instructions here. I enjoyed B most because it was more straightforward and it was just, like, do this, do this do this...and that may seem boring to some people but to me it helps me understand it."
What is 'frustrating fun?' When does something become too frustrating to be fun anymore?
"Problem B is way more frustrating to me than Problem A. In B, you have to find specifics like slope of the tangent line and average rate of change and, for me, if I don't know that one thing I get frustrated and just shut down. In Problem A there is no specific thing you need to find right here and right now, so it's less frustrating because I can feel confident in myself because every individual student is finding, like, their own personal thing compared to B where everyone is trying to find one thing."
"With Problem A, the frustration is with how far can we get. With Problem B, I got frustrated because when I got stuck, I didn't know how to go forward but in Problem A when I got stuck, I just tried something else because there were so many different ways to approach it."
"I think we are all mentioning when you feel like you are falling behind. We are really interactive with our groups and when you talk to someone else and they seem to be getting it and you don't you get frustrated with yourself. With math, there seems to be invisible pressure. Like, nobody is saying 'do it as fast as that person is' but you still feel it. But it depends on the problem because, like you guys are saying, with A you don't feel pressure at all. But with C, I'm not the person to just draw a bunch of triangles so after I drew like 5 triangles and I didn't find anything I kinda gave up...I'm not gonna draw like 6 pages of triangles."
A while back we took a vote and most of the class said they would be happy doing ONLY "POWs." Why is that? What is it about POWs that students like so much?
"I personally love POWs. There are no repetitive formulas that you have to do over and over and over again. I go more in depth with POWs than I would with other types of math problems that just use a formula over and over again."
"I like POWs because I feel extremely intelligent when I'm working on them. When you feel insignificant compared to others around you, you have a tendency to not feel important to the conversation but with POWs I feel like my opinion matters. In terms of college, I think it depends because I don't really feel like I need math for what I want to do in college. As a creative person, I enjoy POWs because they force you to think in creative ways and challenge your brain in ways you have never thought of before. When you have that 'a-ha' moment, there is really nothing like that and I never really experience that moment with Problem B. I mean, I eventually 'got it' and I was, like, now it makes sense but with Problem A I felt like I was doing something bigger than just math. It's almost like 'therapy math' in a way because you just feel really, really good about yourself."
"I really love POWs but I also love the structure and questions of the projects in class. But, like you said, I also love that 'a-ha' moment."
"I also don't really need math for my plans in college. Yes, there is some content that is helpful but with POWs we learn so much more than content. We learn to think outside the box and we have all these 'habits.' Other kids...when they're stuck, they're stuck. With us, we're, like, 'hey, we have other ideas.' We are learning things that, in my opinion, are more powerful than the content."
"There are some kids that really enjoy their content. So, if we were to create a math class like this for all schools I don't know if everyone would like it."
"What if we could have POWs based around content?"
"Well, we already kinda do that. Like the penny problem for instance. It is like one big POW with content built into it. I think all schools should have that."
"It boosts your critical thinking PLUS it gets that math content. And it boosts your confidence and independence."
In your minds, when are we doing 'content' and when are we not?
"I feel like we are always doing content, POWs or not. Like, in C for example, you had to use Pythagorean Theorem to solve for the diagonal. So, you kinda use content in POWs...just not as much as in the Penny of Death problem."
"I went to a traditional middle school and content, for me, was sitting in a class and taking down notes. Then studying for a test and trying to get good grades. But when I came here in 9th grade I had pretty much forgotten all of that content that I learned in middle school. I got a little sad and was, like, 'what did I work for?' and 'what did I gain?' Pretty much just note taking skills...and that's it."
"I feel like we are always learning things, but in here sometimes the content is kinda, like, hidden. Now that we are thinking back on it, I realize I did learn more about the Pythagorean Theorem. I guess sometimes I just don't have an exact name for what I'm learning. You learn something, but then you don't learn it's name....so you might think, I guess we just drew some triangles.
"Sometimes we are just doing work and until points it out you don't realize 'oh yeah, I did content.'
ME: "This is all so interesting to me. To me, content doesn't have to be something that is coming from a textbook or from some source. Anytime we are doing math together and we create a rule...that is content. Doing math together IS the content. It's not like we need to create something that already exists in order for it to be considered a good use of our time."
To me, that is the difference between A and B. In A, because this isn't in a standards list somewhere, I feel OK about just giving you this problem and saying 'let's see what happens.' You all created some interesting rules and those rules are pieces of mathematics that you have created. With B, this is designed to get at a specific thing that does exist. From my perspective, that is more limiting. There is one right thing and, like, if we don't get that one right thing we haven't accomplished the goal."
Do you notice a difference in how your group functions with POWs versus other types of problems?
"I can see a clear shift when we are doing content-based problems versus POWs. There are people who get it and people who don't. Another student and I will still be figuring something out and the other part of our table is like, 'done...we got it.'"
"With content, our table is usually split. But when we are working on POWs, we all work really cohesively. We have all these different ideas that everyone is throwing into the mix and from that 'idea throwing' you come up with this great new idea that we have all created together. And with content it's more like 'this goes to this.' With POWs like Problem A, we had all these different approaches and they were all correct."
"With POWs, even someone is struggling with math will put their idea out there and then we will combine them all together to create this cool hypothesis that we can just go and test. It's really cool to see everyone putting their ideas into this giant pot and we just see what happens."
"I think group work is a lot different when we are doing POWs than content. With content their is one answer and one solution....there isn't really a lot of in between. With POWs everyone usually has something different that we just pile together to create a central answer. I think POWs are a lot better for group work."