How Can a Problem that Feels Incomplete be Finished? | exSTEMsions

How Can a Problem that Feels Incomplete be Finished?

Unexpected answers can feel incomplete

We recently posted a problem where I suspect solvers will get to the end and be wholly unsatisfied by the final answer. Why? Because it’s not a ‘typical’ answer, like a number. The solution to the problem was simply understanding if something is true or not (and why) - yes, or no, with an explanation. Nothing further. I’m sure that some will find this absolutely infuriating. I have presented problems like these to learners of all ages - adults, children, teachers, students, etc. - and inevitably, someone will yell at me: “well, that isn’t an answer!” But it is!  And we need to learn, and to teach others, to be cool with answers that might be different than we expect.

How do you know what answer to expect?

Being cool with an answer that doesn’t feel ‘finished’ is no easy feat. One way to help make sure that the answer you actually get feels satisfactory is to focus very carefully on what the problem is asking you to find. Many times, we make assumptions about the kind of answer we’re going to get, because we aren’t reading carefully or fully analyzing the question. If your problem tells you about an amazing magic rectangle where all of the rows and columns each have a sum of 2017, one might assume that we’ll be finding the numbers that create the magic rectangle. Except, on careful examination, that’s not what the problem requires. The problem requires us to figure out why this magic rectangle MUST be a square, and nothing more. If you understood, very clearly, right up front, that all we need to determine is the shape of the rectangle, then not determining the numbers probably feels much less disappointing.  

Know when it is finished

Another reason to be good with an answer that doesn’t feel complete is that often, problems don’t need us to do all that work! Hello, efficiency! Haven’t you ever done something, and put tons of time and effort in, only to find that it was way more simple than you thought? Weren’t you annoyed that you put in all kinds of effort to do something you didn’t really even need? Of course! If the problem says that we get to be finished at some particular point, then by all means, let’s be finished! Let’s move on to the next challenge, go do something else interesting - but let’s not spin our wheels in a way that isn’t helping us to progress through the problem, just because the answer isn’t the answer we expected. This is not an easy thing to learn to do, because it’s letting go, but it’s worth it!

Don't be afraid to move past the finish line!

Before we conclude, let’s make sure we’re very clear about one thing. Being good with the answer that you have, and accepting that something is finished when you have the answer, even if it’s not the answer you expect, does NOT by any means imply that if you have questions, or things you or your child wonders about, that you can’t explore them. Explore them! When we talk about moving on to the next challenge, it might be something related to a problem that came up as you solved it, or it might be something totally different. All we’re saying is that we should be willing to accept the unexpected - but if you want to take the unexpected, and keep poking at it, and dive deeper, that’s fabulous too.

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