Rethinking How We See Mistakes

I had a flashlight moment recently.

I was helping a student to learn how to balance chemical equations. I had done a few examples, and then I had her try one. Part of the procedure calls for a certain amount of trial and error; you try a number in one part of the equation, then you track how it fans out in the rest of the equation. She hesitated for such a long time, that suddenly I realized what was paralyzing her. She thought she was already supposed to know what the right “guess” was. I told her, put any number, fully expect it to be wrong, and then we’ll use it to get the right one. Her response was immediate, she just put down a number, and it was beautifully wrong, because even though I think she had intended it as a “Here you go I told you how stupid I am” moment of self-pity, she immediately said “Oh wait, no, this would be better.” And it was. It wasn’t right, but it was better. Her wrong answer pointed her toward a better one. It seemed like she had needed permission to jump in with a mistake, before she could even experience the unlocking that happened microseconds later.

This has led me to believe that, at least sometimes, we should be calling those “mistakes” something else. Anything that leads to illumination isn’t a “mis”-anything, it’s progress. So from now on, we’re changing our attitude towards mistakes in my class, starting with what we’re calling them - flashlights. If the growth mindset movement is correct, then what we call things affects how we see them and react to them, both on an emotional level and a cognitive one. I think “flashlight” is more positive, and, more importantly, it’s more accurate. Those flashlights are our guides. They show us where the gaps are between what was taught and what was learned. (read more about these gaps in Dylan Wiliam's “Embedded Formative Assessment”.)

Here are a few other ideas I’ve had about changing my own class's attitude toward mistakes.

Praising

When a student makes a mistake, I’m going to praise them at least as much as when they don’t make a mistake. I want them to know that at that moment, they are straight up legit teaching someone something. I’ve been saying “I’m so glad you said that!” or “That’s the best mistake I’ve seen today!” or “I was hoping someone would make that mistake!” That last one I hope makes them feel like they’re my secret accomplices in teaching. It also creates a sort of suspense in my class, like everyone’s waiting for that magic mistake to happen, the same as if it’s a jackpot they’re all trying to hit. Because that’s what it’s going to feel like when they hit it.

Backtracking

We can legitimize mistakes as learning opportunities if we not only talk about what the mistake was, but where it came from. Because mistakes almost always have some truth in them. For example, when kids distribute incorrectly like this: 3(2×4^x) = 3×2×3×4^x, it helps to say to them – I know why you did that, you were thinking 3(2 + 4^x), which would be 3×2 + 3×4^x. I think it’s a relief for them to know that the way they think has some grain of logic to it, at least enough so that another person can backtrack with them to where attention is needed.

Adjectives

If I HAVE to use the word mistake, then I’ll use an adjective like beautiful, glorious, or brilliant before it, because I don’t want mistake to be a bad word – I want it to be a sign that thinking is happening, neurons are firing, lost souls are finding their way. Those are all beautiful and glorious things to happen in class, and I want as many of them as I can get.

Manners

I’ve also been thanking my students for their mistakes, because they're doing some heavy lifting for us all. For example, the other day I asked if log 3 + log 5 could be replaced with a single logarithm using a log property. One student said no, because they didn’t have the same base. Flashlight! She thought the 3 and the 5 were the bases. Not only did this show me that at least one person was looking for the base in the wrong place, but was also not aware that the unwritten base was 10. Two things learned in one shot because of her, so this was a double flashlight, and I thanked her. Later, another student thought that:

 

would lead to xz = y. Flashlight! At least one person was cross multiplying instead of doing fraction multiplication. They learned when that works, and when it doesn’t, and I learned that I am so not ever going to use cross-multiplication ever again. Thanks, kid!

I’m thinking it would be nice if the other kids thanked them too. I’m not sure if I’ll actually get them to, because that would probably be a bit forced. But the way I see it, the kid making a glorious mistake right away in class, as soon as we’ve done something new, is doing everyone a favour. Everyone else can now avoid making it later, when they’re all alone. If that happens, they’ll either not have any idea that they’ve made one, or they won’t have anyone to help them straighten it out. Much better that it happens when we're all there.

Being not perfect

This is a big one, and it’s probably not going to be popular, even with me. I think our kids need us to not just SAY it’s ok to make mistakes, we need to BE okay with them.

Embracing my own mistakes: Because I’m not just talking about their mistakes, I’m talking about mine too. The ones I’m so careful not to ever let them see me doing. That’s why whenever I have to figure something out in front of them I get so totally flustered that I usually say, “Ahem, well kids, I don’t want to take class time to do this, I’ll ahem figure it out later and get back to you. Move along now, nothing to see here.” And I thereby give them the message “Mistakes are great! For you that is, not me. For me they’re embarrassing, humiliating, and scary and they never happen anyway so yeah.” I need to face it, enlist input, and maybe even get help, for example ask “Why do I doubt my answer is right? What kind of answer would make more sense? Up until where did you get the same thing?”

Problem solving on the fly: When a teacher only ever shows students how to solve a problem that they (the teacher) already know how to solve, that’s great, and of course I do that, but we’re really being disingenuous if it stops there. We’re showing them a nice sequence of rules, being fluidly followed by a calm, confident person who is already in possession of the answer, and who therefore couldn’t be any less like them when they’re solving a new problem. I think we need to give our kids the chance to watch how we handle something that is truly new, something in which we truly have no idea what to do first. And show them how we’re not afraid of that feeling, that nobody needs to be afraid of that feeling, because everybody feels that feeling! Let’s get stopped in our tracks together, try stuff together, mess up, go back, sleep on it, admit we're secretly looking for the answer key...you know, normal life.

Reality

But if a mistake happens during a test, it’s very bad, right? We all know what marks do to kids, and how they absolutely halt all learning, whether the marks are good or bad. So yes, of course, I don’t want mistakes happening then, and nobody does. But they will happen, at any time, so I’d rather get the kids armed with strategies to detect and fix them. And most importantly, remain calm.

New Intro to Log Properties

This went well.

I had thought I'd do this via a desmos activity, and I even started to make the slides, but then decided to do it live in class with everyone totally in sync, because I wanted everyone to witness something at the same time.

Here was the sequence, and the narrative, and the results in italics:

No calculators!

What do your order of operations instincts tell you to do first here:  
multiply the 8 and 16

Okay then do it!

Great, and how much is that log?

7

Some discussion here. How did you get that? Some recognized 128 as a power of 2, some did not. I allowed those who weren't as familiar with the powers of 2 to use their calculator ONLY to try out different powers of 2. NO log buttons. I wanted them to COUNT how many 2's were being multiplied, in order to prepare for more counting AND tallying.

Great! How about this one then?  
Some answered 9 very quickly, some answered log base 2 of 512. We discussed both answers. Those who got the 9 quickly were very good at expressing how they did it by using the already-known exponent for 128 and increasing the total number of 2's by the additional 2. My plan worked!

And what about   ?
16. Lit up the board like a Christmas tree. This was what I wanted. Everyone to just count the total number of 2 factors. I pointed out that now they were seeing numbers through a new filter. Now 128 is 2^7, and 512 is 2^9.

Try these the same way - don't actually multiply the arguments, just see them through a filter: 

I'm happy to report that these were very handily answered and justified correctly, by more than just the usual one or two people.

Now comes the real challenge. Let's go back to that first example, and say it in English, along these lines:

After a bit of coaching:

And what's another way of saying "the exponent that 2 needs in order to get 8"?

After just a hint to use the word log......

Let's go back to the other ones you just worked out and say them in English:

This went super fast!

Now let's generalize:

And for the FIRST time in my career, my students told ME the additive property of logs, instead of the other way around. The third line here was supplied, for the first time, by my students:

That felt good.