June,

It looks like visual in the context of puzzles play a central role in conveying algebra ideas. Is that correct? Can you say more about that?
Thanks,
Arthur

Hi,
In what kinds of settings have you used TTA? Is this being implemented in schools and/or informal settings? Has this been designed to be a full curriculum or is it a supplement?
Thanks,
Elissa

I’d also love to learn more about the contexts in which TTA has been used, and what the project has been learning/exploring about both the short- and long-term effects of exposure to the support curriculum on mathematical habits of mind. Can you tell us a bit more about any studies that might be ongoing or have been completed on this or other student learning outcomes? Thanks!

Are the puzzles delivered on paper or on a computer, and if the latter, are they interactive and what kind of feedback or scaffolding do they provide, if any?

Interesting video.
I am curious about the scripts that the students read to model mathematical discourse: When are they used? Just listening to the video, I can’t quite envision how their use would be natural in the context of the problem-solving orientation of the curriculum.

Thanks, June and Paul, for posting the video. Inspired me to finally order a set!

You mention that students feel more like spending time with a puzzle vs a problem (or exercise?). Are there other specific behaviors that identify when a student is seeing a task as a puzzle? That is, how do I know if I am posing a puzzle or a problem to a student? Back to the old question of defining what is a math problem – problems for some are simply exercises for others. Is the same thing true of puzzles – if not, why not>

Hi Gerald. We’d love to know more about this ourselves. When we use the words “problem” and “exercise,” we’re pretty clear about what we mean by the difference: an exercise is just moving the weights around, and a problem requires some thinking. Between a “problem” and a “puzzle,” the difference is, as you point out, not so crisp, and depends on how the student sees the task. That depends partly on the kid (of course) and partly on the nature of the task (of course) and partly on how “we” (curriculum and/or teacher) present it. I’m mindful of the Brown & Day (2006, I think) study of stereotype threat in which the same task (Raven’s Progressive Matrices) was presented with all the usual instructions but framed, in advance, either as an IQ test or as a set of puzzles and the outcomes were quite different. Some “problems” (e.g., Mary paints a room in 3 hours and David paints it in 5, so how long would it take if they work together?) /are/ just puzzles, with not a shred of real life validity to them, but are presented as “problems.” I guess the real difference—the behavior that identifies when a student is seeing a task as a puzzle—is the more patient-with-self, try-and-see approach. I’m supposed to be able to solve problems, and if I can’t, um, maybe I’m not so good. I’m supposed to be puzzled by puzzles—that’s what makes them puzzles—so if I’m puzzled, well, fine, I can just figure it out. /NOT/ everyone likes puzzles, but enough people do to keep supermarkets (not just high-end bookstores for geeks) selling them. The characteristic of a puzzle is that you’re quite sure at the outset that you /can/ solve it, if you can just find some way to start. That’s how we’d like our students to handle all of the work they’re doing, and one of the (several) reasons why we like infiltrating puzzles throughout, making them not a dessert but a part of the main course. Ideally, I’d love teaching to treat even the non-puzzliest routine stuff with a bit less existential weight than we sometimes see. For a particularly mundane example, confusing 7×8 with 6×9 is /so/ routine an error that it can be treated as funny rather than tragic /even/ if we really care that the kid sort it out sooner rather than later. And the lighter touch might actually speed the sorting out!

Pardon the slight digression. I think it’s only slight! :-)

Your presentation indicated that your materials were used as a supplement the regular algebra course. Who teaches each? Is there coordination of approaches? How has that worked out?

Ah, yes. The question! :-) That has varied a whole lot. While we were doing the development and field tests (prepublication), one school had a “math lab” that was neither contiguous with the algebra class nor taught by the students’ algebra teachers, /nor/ were the teachers who taught TTA necessarily teaching /any/ algebra class that year. And no explicit coordination (though teachers knew each other, of course, and sometimes talked informally). In another school, all the TTA classes were conducted during the last block of the day, and all of them by the algebra faculty, but students did not necessarily have the /same/ teacher for TTA as for algebra. In one middle school that is using TTA to launch kids into algebra early (including their weakest students), it is not a parallel course, but a component of the one mathematics course the students receive during the year. All that I can easily say about how it “works out” is that I have to believe (gut dictating belief!) that coordination would bring more benefits, but lacking that coordination doesn’t seem to have worked mischief. We heard a few times Algebra Class teachers expressing pleasant surprise when their “at-risk” students (the ones in TTA) found polynomial multiplication particularly easy (partly because our visual area model helps them, but possibly also because they’d actually encountered it in TTA /before/ they reached it in Algebra class, so they were ahead of the game). It /could/ be (but understand, please, that I’m making this up!) that the surprise for the algebra teacher turned into a boon for the kids by raising their impressions of what the kids could do, a benefit on top of the actual logical learning that the kids got from TTA. We don’t know, though. The implementations varied, as did the amount of information we were able to track in each school.

I love this puzzle idea! Is there something about a puzzle that seems more like an invitation to stretch your thinking rather than a requirement to find an answer? Could that sense of puzzle — invitation rather than requirement — make a difference for students? Or is there a difference in the very type of activity in a puzzle versus a problem? Or is it a bit of both?

Hi Sybilla, Sometimes what is attractive about a puzzle is that there’s not the expectation that there is one answer but for example in a Sudoku puzzle you need to find a starting point, figure out some information, and use that information to help you progress in making sense of other parts of the puzzle. We want students to use these same dispositions to solve mathematical puzzles, and in the curriculum, we try to pose problems/puzzles worthy of that kind of reasoning and strategic thinking. We also want students to bring these kinds of habits to solving more typical mathematical problems as well as puzzles. thanks, june