## Pick one piece of mathematics that you are proud of. Why did you pick it?

“I really like to work with conjecture and testing stuff so I liked when we were able to experiment with trying to get certain numbers using consecutive sums. I like being able to be ‘hands on’ with stuff and try things out to see if they work or not.”

“it made sense”

"We went farther than the original problem and it was really fun doing that."

"at first I wasn’t really sure what to do with it but then when we had time to collaborate with our group, my group member had this really good idea and I was able to build off her idea to create the rule we were looking for. Also, I really like that at the end of class we had the opportunity to keep pursuing it in different ways by looking at challenge options. I thought that was fun because it let us go beyond and I was able to come up with a different rule for a bigger problem."

"I saw patterns going on with it and then I found a rule for it. At first I just based my rule off the pattern that I saw but then I understood where the rule came from and which squares it was talking about."

"I went about finding a rule in a different way than a lot of other people did but I still came to the same conclusion which I thought was really cool and it was good for me to be able to explain the way I do things. Our group also worked on an extension problem and we actually came to part of an equation for that.”

"seemed more like a puzzle than a traditional math problem so I experimented a lot more with it. It was something I could play with."

"testing out what works and what doesn’t and explore."

Pick a piece of mathematics that you were NOT proud of. Why did you pick it?

"we didn’t spend enough time on it in class"

"we had these restrictions that seemed kinda confusing. I think it would have been easier if we had some sort of freedom"

"frustrating and it seemed like this little thing we did for like a day and then it disappeared off the map.”

"I was confused but other people got it"

"It didn’t connect to anything. It just kinda ended…there was nothing to explore"

"it seemed like an enormous problem, like it could go on forever and there was no sure way of knowing."

"a person at my table gave me a rule for the problem but I didn’t do any of the work for it and I didn’t understand it because it was someone else’s work."

We start each day with one big, open question and then use the class period to explore that prompt. How do you experience these questions and the idea of experimenting and playing?"

"I like it because it is usually a question that can be answered in a bunch of different ways so every time you hear from somebody it’s a different answer and it makes you think."

"You have to think about it a bit more on your own and develop your own ideas first."

"It helps you because you can create your own ideas for how to solve that problem. Then we you talk in groups or with the whole class you can bounce ideas off of everyone and they can give you their feedback and then you end up with a rule or a way to solve the problem that is even better than what you could have come up with on your own."

"We have freedom and, in math, you usually don’t think of having freedom. I used to think of math as having a bunch of rules and everything was set and strict. With our open questions, it just kind of blows that concept away. They make you stop and think instead of just memorizing a rule and that will help us outside the classroom or when we have a problem with a rule we don’t know."

"It’s a really nice way to get away from traditional math where it’s like, ‘here’s a worksheet, go figure it out…if you don’t get it, then tough.’ I think experimenting and playing with our questions is a great way to get your mind going."

"I don’t mind them but they frustrate me after a while sometimes because I wish there was just one answer to get to rather than something that just keeps getting bigger and bigger. Sometimes it just keeps going and I lose interest and just want to move on to the next problem. I think it might also be because my previous education was very traditional and so I have become used to just having one answer."

Another student's response to the above quote:

"I have the same problem where I get bored easily, but I like it in here because our problem kinda changes when we look at extensions."

I find these focus groups fascinating. Our students have such incredible insight into the intricacies of the classroom. I was struck by the themes present in work that students were NOT proud of. They noted they didn't connect to work when:

1) there wasn't enough time to let them develop ideas,

2) they didn't see the connection to a larger theme/unit/purpose,

3) there was not an appropriate level of difficulty, and

4) when the problem didn't allow them to think in a way that was natural to them (no room to explore or confusing restrictions).

On the other hand, it seems much easier to boil down their thoughts about work that they DID connect with. As I listened to them and re-read their responses, I just kept thinking, "they are telling me to just let them explore an idea and develop their own ways of thinking and understanding." The discussion about 'real world' comes up a lot in the math ed community. I have seen students be incredibly motivated by exploring block patterns and drawing shapes and I have also seen students be incredibly bored by problems that are based in some real world application. Mostly, I just think students want a problem that does not seem unnatural to them and one that allows them to explore, think, and develop their own 'wonderful ideas.' As Eleanor Duckworth writes:

"Textbooks and standardized tests, as well as many teacher education and curriculum programs, feed into this belief that there is one best way of understanding, and that there is one best, clearest way of explaining this way of understanding."

"...there <are> a vast array of very different adequate ways that people come to their understanding."

"...children can probably be left to their own devices in coming to understand these notions."

*could*be helpful is to broaden our mathematical goals for students. I'll use the current work with my students as an example. We are currently working on a unit in which we are investigating the area of the Koch Snowflake. In the past, I have used that as an opportunity to engage students in investigations with arithmetic and geometric sequences (part of our Algebra II standards). This year, I broadened that goal to engage students in "looking for patterns and making predictions." That broader goal has allowed me to let students pick multiple paths and explore things that are interesting to them...so long as they are looking for patterns and making predictions from those patterns. I feel confident that the students this year would be able to work with arithmetic and geometric sequences as well as, if not better than, my students in the past even though, this year, we never even discussed them specifically.

There is still a lot to think about, and good task design does not necessarily ensure a rich mathematical experience for students. But, I think it is a good start.