Well, if you ever come up here, I expect to feed you dinner and have you go through my kids’ math assignments! Most of what you say goes OVER my head, because I focus a lot more on the Developmental Math prior to Algebra with my kids. But I still learn a lot from your posts.

it’s possible to take problems that have been fully abstracted by a textbook… and then facilitate that abstraction with our students. But I think we should expect curricula to help teachers facilitate that abstraction

I see that as you are coming to make meaning of this concept abstraction, it is a notion that is defined within a mind-free mathematics, not within the realm of people’s ways of knowing. That may be a way to think productively of curriculum authoring.

@bmeyer
It strikes me that you would say the curriculum could not be named until after-the fact. I think this reflects the constructivist’s view on curriculum (c.f. Steffe, 1990, “Mathematics curriculum design: A constructivist’s perspective”).

@michael p
My first thought for getting beyond the printed text is to lose that constraint that it should serve as both a classroom tool and a reference. Similar to what I think Dan imagines, I tentatively suggest that the ideal textbook would be a sequence of mathematical problems that follow a research-based trajectory of student learning. Of course, this trajectory does not match the potential for every student, but has shown to be viable in many classrooms.
Leaving print, could allow for the teacher to sequence the problems posed based on the last lesson — a “choose your-own adventure” story if you will. Some options, but still structured. More powerful would be if these sequence of prompts were held together by a powerful storyline, or even more ideally a larger problem–what many define as a problem-based curriculum.
While Dan Meyers task ideas might effectively engage kids on a Friday, he makes quite clear that in his view, teacher’s should not be asked to create curriculum; an unwieldy job–as I am sure he is learning. I am uncertain that one person’s text or image-based or video-based problems would be that of another, whether it be enactable by all teachers, or strike a chord with all learners.
I suspect Polya, who greatly valued problem-posing, recognized this dilemma when he began to focus on encouraging questioning. Dan Meyer recognizes this as well. Maybe a good text is characterized by the potential to generate worthwhile questions in learners? I know Paulo Freire, Ubi D’Ambrosio, and Rico Gutstein value precisely this quality in mathematics education.

@Brian

Curriculum materials are designed to be usable by as many people as possible. If they are too difficult to use, then people will not use them, they will not be bought, and the company producing the materials will hurt financially.</blockquote.
Sigh, the words of a corporation, not of an educator.

in bend
brooke

I have the benefit of working at a school where students don’t use textbooks and teachers are curriculum designers for their classes. I know that not everyone is in the same position. But, it seems that improving curricular materials is only a small part of improving teaching.

I’m glad you’re happy at your job but I don’t think we should expect every teacher to teach without curricula. Their efforts would be incoherent and duplicative in many cases. The Exeter and Park Math curricula makes us all better off, etc.

And absolutely, it’s possible to take problems that have been fully abstracted by a textbook (or not abstracted at all, in your example) and then facilitate that abstraction with our students. But I think we should expect curricula to help teachers facilitate that abstraction or, at the very least, not hinder it.

Michael Pershan points out that the cell phone problem isn’t intended for an interaction between a teacher and a student. It’s in the practice set.

He’s right about that, but the material that’s meant for student / teacher interaction looks exactly the same only with all the steps worked out. Even in the student edition, the worked examples that precede the practice problems start at a single (high) rung on the ladder. The only difference is they show you how to perform all the operations at that rung.

I don’t think textbooks should exclude those worked examples, FWIW. But I want to see a) more rungs, and b) the opportunity for a student to do something at every rung.

Michael seems to be envisioning some kind of feedback system that helps the student up the ladder. I don’t think computers are good enough for that kind of help yet. The teacher should be that help. What computers can do is just show a context, ask a question, and then aggregate the class’ responses for the teacher. The teacher can do something with those responses and then tell the computer to move up to a higher level of abstraction. Repeat as desired.

Here’s a text-based version of the cell phone tower problem that, I claim, does not already do the abstracting for the students:

Three towns are buying a cell phone tower, but arguing over where to place it. Where is the fairest place to put the tower?

This problem, as stated, is wicked hard. Absolutely none of the abstraction has been done for the student. Now, we could put this in front of a kid, but we won’t. They’d freak out, have no idea what to do, throw a fit, etc. So we want to give them more guidance.

The issue isn’t just “How do you get text-based curricula to leave off the abstraction?” It’s more like, “How do you provide some guidance towards abstraction without giving the whole thing away?”

I think it’s worth teasing this out because text-based curricula have another option. They could systematically provide students with the tools for moving themselves up and down the ladder of abstraction. When students encounter a problem that offers no abstraction they can learn to try various things: taking guesses, finding different ways to visualize the scenario, tinkering, being playful with notation, etc.

For various reasons that might not be a live option. But I think it’s true that print gets into trouble because kids can’t move up and down the ladder themselves. But there are strategies for negotiating the ladder, and kids might spend time learning them.

Here’s a challenge: The lesson above helps a student move up and down the ladder of abstraction for the cell phone tower problem. What reasons do we have to believe that this student will be better at moving up and down the ladder on their own? (Is it just because we’re explicit when we make the moves up and down the ladder?)

My understanding — correct me if I’m wrong here — is that a fundamental constraint on textbooks is that they should serve both as tools for the classroom and references for later. These texts are supposed to make sense to a kid who is studying them on her own. The homework problems should also make sense to a kid on her own.

If what we’re talking about here is classroom use of curricula, then the problem isn’t print, because no one is forcing teachers to toss up Q21 up on the board during class time. Sure, there are better things for the teacher to be doing, but that’s hardly print’s fault. It’s more, like, you want curricula to provide things other than print, like slide decks and videos.

But how do you see digital curricula transforming a question like the cell-phone one for homework or assessments? Do you want to provide narration to your pictures and state the problem in a video? How big of an improvement, LOA-wise, would that be over the print version? (That last question is sincere, not rhetorical.)

I hope this will not lead to digital content obesity.

For an interesting experiment in how to abstract in a way that makes the thought process visible, I’ve been enjoying the blog What If, by the people who brought you XKCD.