What Mistake did I Make?

My kids are back in school. In recent years when we go to parent-teacher conferences and back to school nights I have offered to be a guest speaker, but no one has taken me up on it yet. This year, it seems like my daughter’s fourth grade teacher will, and I offered to speak on the topic of problem solving. The teacher was excited and commented that she’d let the choice of subject be a surprise since problem solving is applicable to everything.

When I speak about problem solving, it is always through the same lens as this blog that you often can choose which problem to solve. The next day I mulled over how to teach that concept to a bunch of 9 and 10 year olds. Usually, as you have maybe noticed in this blog, I rely on examples of decision making that my audience will have experience with. Choosing what to eat for dinner, what car / appliance to buy, etc. (Un)fortunately, I think the average 9 or 10 year old does not have that much autonomy. And even if they did, I’ve gotten the critique from a friend speaking for other adults that “most people do not think about what to eat the way you do Zohar.”

So, my go-to approach is not an option. I had a 3-hour drive with my kids and so I tried a few ideas. What I landed on is flipping a common scenario. Say you’ve solved a problem, and for whatever reason, you think your answer might be wrong. How do you work backwards to identify the error? My sixth grader confirmed that such a thing had never been taught to them before. He also confirmed he had faced that problem when say, working on a multiple-choice problem and he got an answer that was not one of the options.

Ok, so I want to teach a methodology for figuring out what problem you did solve when you learn you didn’t solve the right one. First, we can think about the kinds of errors that could be the source of our suspicions:

• I might think my answer is wrong because I was not sure how to solve the problem in the first place.

• I might think my answer is wrong because I know I have a habit of making small numerical errors.

• I might think my answer is wrong because another source has a different answer.

• I might think my answer is wrong because it doesn’t seem to make sense.

When you are in this situation, you have a bit of a debate on your hands. Should you put more work in to try to get an answer you have more confidence in, or not? Well, it depends on the stakes as well as your ability to put more or different effort in.

For standardized testing, students are taught strategies to make sure they get through all the questions well, and to save some time if they have extra to revisit the harder questions. My kids also have some adaptive computer-based testing. If you get a question wrong you cannot go back, and in fact the rest of the exam will be easier than if you had gotten it right. So, the stakes and your available capacity can vary quite a bit.

If you do have the capacity and it’s worth trying again, you can use very different strategies depending on what situation you are in. I tend to make a lot of small numerical errors, and so I learned that my answers always have a decent chance of having an error like that in them. To combat that tendency, I do a lot of validation through estimation or guess-and-check. (Side note, this feature made me look like a wizard when I tutored students in college. They would get a random wrong answer, and I would immediately point out the one step out of four they got wrong. Because I had just made that mistake and corrected it while they were working through the math).

If on the other hand you think your answer might be wrong because your response is not one of the multiple choice answers, you can conclude that you made a non-obvious mistake (because the obvious mistakes are usually the distractors AKA wrong answers). Most often for me that looks like leaving out / including something in the problem statement that they intended for you to do the opposite with. This kind of error also shows up online as meme brain-teasers. “95% of people won’t notice that when you have this whole crazy arithmetic problem that’s multiplied by zero, the result is zero.”

A different version of disagreeing with another source is when two students are solving the same problem and get different answers. Hopefully if you both agree then the answer is actually right. But if you don’t, there’s an added layer of who is right and what the wrong person/people did differently than intended. As I joked to my oldest, the answer is not just “Whoever has the higher test scores is right.” But walking through the two solutions you might easily see where someone went wrong.

For the other two categories of possible mistakes, I find it helpful to think of alternative ways to solve the problem. Typically, redoing the same math will get you roughly the same answer, so if the answer doesn’t make sense or you are not sure you even tried the right thing, you can try a different method and see if you get something similar. A great illustration of this is Galileo's Leaning Tower of Pisa experiment. According to the story, if you drop two things that weigh different amounts, they’ll still land at the same time. However, if you drop a sheet of paper as well as a crumpled paper, it takes very different amounts of time. Not because gravity is wrong, but because there are other factors in play (wind resistance) that change the results.

I am feeling much more optimistic about how this all might land with a bunch of fourth graders. I can also think about building an activity to diagnose some wrong answers to a problem. Suggestions welcome!