Yowza. Solid PD outline.

For me – it’s about sequencing tasks to mentor student mathematicians (it’s not about discovering, it’s about building mental relationships they can use to solve problems) and it’s about sequencing PD tasks to mentor teachers.
Dan’s sequence in this example is worth studying.

Hi Vince, thanks for the note. I’d love to help out with your PD efforts. Frankly the only work I’ve ever done that seems to resonate with teachers has involved the same three-step process every time: a) do weird math together, b) debrief the math to decide what made it weird, c) decide what we’d need in order to start being weird in our own practice.

Do you have recommendations for sharing this idea of intellectual need with more teachers? I provide professional development for area schools and would love to share this message with teachers who do not attend conferences or don’t read your blog. Do you have examples of teachers creating intellectual need so that I can inspire more teachers?

Thanks for your thoughts here, Scott. Your comment brings to mind a session of yours at CMC North that expanded on these ideas significantly. I need to revisit my notes.

This is super helpful analysis, Sue. Thanks for sharing.

Perhaps the most inviting reason to use surprises is to get students to pursue the question you want them to consider. Everyone enjoys a puzzle, IF they think that they can figure it out and IF they then are successful. People like to feel clever. And feeling clever builds mathematical confidence.
It isn’t so easy to find problems that create useful surprises, but the more you use them, the more students respond. I want students to gradually build their belief that they CAN figure out whatever weird thing they encounter in my classroom. There’s a crucial unspoken agreement here that I learned from Rick West — I won’t give them a puzzle that they don’t have the means to solve. I want students to progress to the point where they see the surprise and then say something like, “Don’t tell me, I want to figure this out.” You can watch individual students progress in their mathematical confidence in the way that they respond to surprises. I do think that this is related to having a growth mindset.

Jan might be someone whose prior experience with puzzles has provided her with some confidence, some belief that this surprise was worth pursuing, even though it was outside of her area of subject matter confidence. The set-up to the problem might have worried her that the talk had prerequisites that she didn’t meet, but the puzzle drew her in.

One more thought here: I would like to try the same set-up to the problem (perhaps they should choose a whole number between 0 and 4) and go through the same process, except have the students plug their chosen numbers into the polynomial that is the expanded form of
x(x-1)(x-2)(x-3)(x-4).
For students at just the right place in their understanding of polynomials, this could lead nicely to an investigation of the relationship between zeroes and linear factors of polynomials.

You’re right.

We can’t under-value either, because they are not mutually exclusive. But let us assume for argument’s sake that they are. Would a teacher be better off devoid of sound pedagogy and yet solid psychology, or vice versa?

I could take a group to the moon without pedagogy, but would sink without psychology.

Chester, I sometimes explicitly say something like, “I don’t expect you to understand this particular explanation right now. It’s based on calculus, which you’ll be formally meeting in a later class. But I’m offering it so that you know that it does work, that there’s reasoning behind these mechanisms.”

I think you and I are probably closer to being on the same page then we’re acting. I had some excellent teachers in my day that explained mechanisms, and I had some teachers who just taught the mechanisms.

I find it hard to believe that what changed is the belief in her capacity to solve this problem.

Love this line.

It seems she did not necessarily overcome any anxiety or aversion, but rather temporarily disregarded it in pursuit of something worthwhile. This creates a portal, allowing you or her a crack at building a growth mindset, ability, etc.

I think pedagogy is the science of generating and inviting as many such moments as possible.