Open ended, open middle

At last! I found the phrase! So often people talk about "open-ended" problems when they are talking about math puzzlers with exactly one solution. That's just not right. I adopted the language "problem-solving problems" but that's just awkward. But finally, some genius came up with the right phrase, the one I've been looking for all these years: Open Middle. That's it!

In my math-ed classes in grad school, they touted the benefit of open-ended problems. Like this one:

James needs to mow his lawn. The lawn is a rectangle, 12 feet by 16 feet. He needs to start and end in the top left corner, and he doesn't want to push the mower any longer than necessary to get the job done. The mower mows a path 14 inches wide. Show one path that James could take to mow his lawn that involves as little overlap as possible. 

I can imaging students having different good answers to this problem, with 3 or 4 "best paths" emerging in the end. Students might argue about whether or not turning the mower around a corner is more work, and some might talk about the need to overlap the path a little to make sure you got the edges fully. The problem is clearly stated, but no direction given for where to start, and multiple correct answers might come up.

Here is another problem that was also presented as "open-ended" in that same class.

A simplex lock is a lock with five buttons, numbered 1 - 5. You can set whatever combination you like, with the following constraints: 

• You need to have at least one button pushed in your combo, but you don't need to necessarily push all 5. 
• Each button can be pressed at most once. Once it is pressed, it stays down. 
• The order you press the numbers in matters. However, you can also press two or more numbers at the same time. So, 1 - 2 - 3 -4  is a different combo from 1 - 3 - 2 - 4, and it is different from 1 - 2/3 - 4 (where 2 and 3 get pressed together). But 1 - 2/3 - 4  is the same as 1 - 3/2 - 4. 

How many possible combinations are there? 

This is an interesting problem (to me at least). It is not obvious where to start, and I never learned a method for solving a combinatorics problem of this type before. I can imagine many possible ways to try to solve it - mostly breaking the solution space into "types" of combinations and then counting the combos in each type. For example, you might look at all the combinations with exactly 1 button, then those with exactly 2, then 3, etc. But different students might break it down along different lines, leading to rich discussions about what methods work best and what range of methods we could use.

But open-ended? Not quite. There is exactly one solution, one final number of possible combinations. In fact, this would be a great problem to bring up (if it weren't so complicated) when people tell you that math is always right or wrong, 2 + 2 equals 4 every time. Yes, it is true, math often has correct answers. But don't tell me math is straightforward. Here is a problem that has a single right answer, but finding it is anything but simple, and memorizing a procedure to solve it is just crazy. (Interesting conversation about that here.)

What happens when we talk about these as open-ended problems when they are not? Maybe we don't give actual open-ended problems a fair shake. Most of the actual open-ended problems I've come across involve modeling, perhaps with statistics, often making assumptions or estimations. This is such a valuable skill to have, and it leads to multiple "correct" answers. We need to be teaching more modeling problems, and let kids know that there is not always a single correct answer when they are using their math brains to do a math problem.

And I need to way to communicate with parents about problems that involve multiple paths, about the joy of problem-solving, about the creativity that comes about when you let your kids loose on a deep, complex (for them), problem with no clear path to the solution. "Open middle" describes it perfectly. The problem is stated clearly, and there is an answer to be found, but it is not at all prescribed how to get from here to there. You have options, and the process belongs to the the student, not to the teacher.

This is also a way to combat the idea that all math processes that come up in the classroom should be taught.

Because there are many that CAN be taught, and some classes teach them, but I don't think it does anyone any good. But that's for another blog post.