A couple weeks ago, I participated in a workshop with Robert Kaplinsky. He was talking about "Open Middle" problems. These were basically a typical math problem, like 8-3x=12, but with blanks instead of numbers. The questions come in three parts that ask the students to fill in each blank with a number between one and nine, with some condition attached. Conditions range from "so that the solution is equal to zero" to "so that the solution is a small as possible" with other conditions between. Check out his website for more details, https://robertkaplinsky.com/ He can explain it far better than I can.
I'm thinking about these type of problems, because they remind me a bit of those old "Choose Your Own Adventure" books. The reader takes on the role of the protagonist and starts with the first chapter, but at the end of that, you, the character, are offered a choice of actions. Depending on which choice you pick, the book tells which chapter to read next. There's another choice at the end of that chapter, and so on until you find the treasure, or rescue the prince, or die a horrible death at the bottom of a spiked pit. The great thing was, even if you reached the horrible end of the story, you could back up and try again. I read some of those books over and over, making different choices each time, just to see what would happen. (And isn't that what problem solving involves?)
The Open Middle problems are like that because once you get past the first part, the problem opens up with lots of choices in the middle. Students can engage at a number of levels by guessing and checking, making a table, thinking about a general rule, or whatever approach they think of. Then, the students get to tell their part of the story in class! My job as the teacher is to see which paths everyone is taking and get a conversation going about those choices. Kind of like when my brother read the same adventure book I did, and we talked about the different choices we made and the endings we found. The other nice thing about Kaplinsky's problems are that it's okay if students don't get to the third part. It's designed to be a harder question, with multiple ways to solve, but unlike the middle, does not have many correct answers. (At least for the ones I've seen so far.) I'm thinking that after the students work on the first two parts, and discuss their methods, I would help solidify the math idea, and assign the third part for homework. I certainly wouldn't grade it on correctness, as some of these problems are hard. But I would try to provide feedback to the students about whatever they write or tell me about their process.
I'm looking forward to digging into Kaplinsky's work some more and trying these problems in class. It probably won't go well the first time, but the fun thing is, I can make a different choice and try again the next day. Math is nowhere near as dangerous as that spiked pit.
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