Week 1: Chapters 1 - 3, Question #3 When students encounter new ideas that do not appear to fit into their existing mental model...

Eric-

Thank you for your post.  My copy of the book has not yet arrived, but I'll respond anyway.

> "p. 12 “'Every time a student makes a mistake, they grow a synapse.' I've heard Jo Boaler make this claim before and to be honest, I've been pretty skeptical."

I definitely agree that I do not believe this either, so I was compelled to go hunt down the research.  The study Boaler cites is Moser 2011 [http://cpl.psy.msu.edu/wp-content/uploads/2011/12/Moser_Schroder_Moran_et-al_Mind-your-errors-2011.pdf].  In the conclusion of this study the authors state, "The findings reported here are consistent with previous results demonstrating that growth mind-sets are associated with adaptive responses to mistakes (Dweck, 1999, 2006). Specifically, a growth mind-set was associated with enhanced Pe amplitude—a brain signal reflecting conscious attention allocation to mistakes— and improved subsequent performance. That the Pe mediated the relationship between mind-set and posterror performance further underscores its significance in linking mind-set to rebounding from mistakes."  (The Moser article does not use the word "synapse" at all.)

But yes regardless, I still think that it is helpful for our students to hear that making mistakes help them learn, especially with how mistake-adverse so many of our students are!  It takes so much work to help them overcome that.  I don't want to go so far as to lie to students by telling them that they're adding a synapse with every mistake, but depending on the audience, simplifying the message down to "mistakes grow your brain" can make the idea more believable/acceptable to students.

>"I want explore what it means to make a mistake. I'm guessing we're talking about conceptual misunderstandings, not computational errors."

I think based on Moser's study, a student with a growth mindset who makes a mistake would still have increased Pe amplitude and therefore more conscious attention to his/her mistakes.  A computational error can come when students are rushing/feeling anxious/press the wrong button on the calculator/etc./etc. but perhaps making that mistake will cause the student to take the time to not make the error again on subsequent problems.  Moser: "Pe amplitudes are associated with adaptive behavioral adjustments, such as slower and more accurate responses following mistakes."

>"I think we need to present students with mathematical situations that allow them to struggle and be confused potentially before they figure something out to resolve the confusion. Designing this well is easier said than done, I think."

I agree! I'm hopeful for more discussion on productive struggle during this book study.

Jennette

"Mistake is probably the wrong word for explorations and blind alleys, assuming we’re working on a math problem which doesn’t have an obvious procedure to be solved. It’s really the struggle, the cognitive disequilibrium, that causes the increased electrical activity, right?"

I think so. In the book, Jo Boaler says,

"When I have told teachers that mistakes cause your brain to grow, they have said, "Surely this happens only if students correct their mistakes and go on to solve the problem." But this is not the case. In fact, Moser's study shows us that we don't even have to be aware we have made a mistake for brain sparks to occur. When teachers ask me how this is possible, I tell them that the best thinking we have on this now is that the brain sparks and grows when we make a mistake, even if we are not aware of it, because it is a time of struggle; the brain is challenged, and this is the time when the brain grows the most."

So it seems like it's not the mistake but the struggle that we assume a student experienced if they made a mistake. But what if they struggled and got it right? Or what if they didn't struggle and made a mistake?

Eric, I really like your idea of unpacking what we mean by a mistake. It is helpful to think about the different ways I use the word and what it means. And what it means to students. 

One important aspect of mistakes in learning for me is about risk taking.

I often share a fable with students that I was obsessed with when I was a kid. It comes from Leo Tolstoy.

The Duck and the Moon

A duck was once swimming along the river looking for fish. The whole day passed without her finding a single one. When night came she saw the moon reflected on the water, and thinking it was a fish she dove down to catch it. The other ducks saw her, and they all made fun of her. From that day the duck was so ashamed and so timid, that even when she did see a fish under water she would not try to catch it, and before long she died of hunger.

I really like it because there is a lot that we can all relate to and it is a good frame to for students to talk about classroom community and how people learn. 

And risk taking goes both ways. I used to record the audio for my classes and I remember a long time ago there was one class where a student had answered a question incorrectly and I called on another student. It really upset me for a long time. I had moved on from the student who was wrong because I was afraid of making them feel bad. The question I would ask that student today is "How did you get that?" (which is the question I ask every student who gives me an answer, right or wrong). But at the time I was afraid of that question and of my ability to handle their answer. What I realized at the time was by moving on from that student I had conveyed to the whole class that mistakes were to be feared, avoided and ignored. I'm not sure if my brain grew as I was making that mistake, but it grew a whole lot from thinking about it.

Another aspect of mistakes is feeling comfortable with uncertainty. 

I use guess and check often as a problem-solving strategy. Asking myself "What could it be?" and then testing that possible answer is something I can almost always do when I'm not sure how to start. It's often something I do for a while and then stop and look back over my work to look for patterns. I do a lot of work with students to first appreciate and then to develop this strategy. There is often a lot of understanding that goes into being able to check an answer. 

Here's a problem I love to use with students to draw out student problem-solving strategies:

Farmer Montague raises chickens and goats. She is not sure how many she has of each animal, but she does know that she has 22 animals altogether. She also knows that, altogether, her animals have 56 feet.

How many of each type of animal does Farmer Montague have?

One of the questions I ask students who are struggling to get started is, "How many chickens could there be?" For some that is enough to begin and start playing around. 

For others, it can go like this:

Student: I have no idea. 

Me: Can there be 100 chickens?

Student: No. 

Me: Why not? 

Student: Because there are only 22 animals. 

Me: Ok, so how many chickens could there be? 

Student: But how am I supposed to know how many there are?

I hadn't thought about this exchange in the context of mistakes before, but I think it really is rooted in this idea that math is a "you either know it or you don't" subject and a fear of being wrong. Guess and check is a great problem-solving strategy, and it is also a may to help students see mistakes as productive, which I think is an important part of helping students feel comfortable making them.

Kudos to all who posted here  - you made this sleepy brain think. Perhaps Boaler has overstated when she says every time a student makes a mistake, a synapse grows or maybe, as some believe, success most often reinforces superstition. I once had a puppy that got into the garbage can for what seemed the hundredth time.  In frustration, I took his paws and tapped them on the garbage can, repeating, "No, no, no!"  A few moments later, his head was back in the garbage can, but his paws were painstakingly curled under so as not to touch the can. Dogs sometimes REALLY know how to teach!